Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Zhang, Xin
Master of Applied Science - MASc
University of British Columbia
The sum of the lognormal distributions is widely used for performance analysis in various areas, including wireless communications, free-space optical communications, finance, etc. However, the lognormal sum probability distribution function does not have a closed-form expression. In this thesis, we comprehensively survey various approximating methods of the lognormal sum and investigate some related practical applications. Based on the survey, we derive a new approximation model using the Marcum Q-function having higher accuracy than the existing approximation approach.
A Survey of the Sum of LognormalDistribution and Some Recent ResultsbyXin ZhangB.Eng., Tianjin University of Technology, P. R. China, 2006M.Sc., Xidian University, P. R. China, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2022© Xin Zhang, 2022The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis entitled:A Survey of the Sum of Lognormal Distribution and SomeRecent Resultssubmitted by Xin Zhang in partial fulfilment of the requirements of thedegree of Master of Applied ScienceJulian Cheng, School of EngineeringSupervisorChen Feng, School of EngineeringSupervisory Committee MemberJahangir Hossain, School of EngineeringSupervisory Committee MemberYang Cao, School of EngineeringUniversity ExamineriiAbstractThe sum of the lognormal distributions is widely used for performanceanalysis in various areas, including wireless communications, free-space op-tical communications, finance, etc. However, the lognormal sum probabilitydistribution function does not have a closed-form expression. In this thesis,we comprehensively survey various approximating methods of the lognor-mal sum and investigate some related practical applications. Based on thesurvey, we derive a new approximation model using the Marcum Q-functionhaving higher accuracy than the existing approximation approach.iiiLay SummaryIn wireless communications, propagating electromagnetic waves are sub-ject to attenuation and fading when the waves pass through obstacles. Phys-ical reasoning can be used to derive the statistics of the signal attenuationas a lognormal distribution. This thesis proposes a method to approximatethe lognormal sum more accurately.ivPrefaceAll the work in this thesis was finished under the supervision of Prof.Julian Cheng. I was responsible for the survey of the sum of the lognormaldistribution. Based on the current research in this area, I derived a new geo-metric method, and mathematical analysis and simulations were performed.Julian Cheng was involved in selecting the study field and revising the re-sults and the material. We are also planning to publish the material of thisthesis in IEEE Communications Letters.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . 11.2 Lognormal RV and Overview of Lognormal Sum Literature . 21.3 Thesis Organization and Contributions . . . . . . . . . . . . . 5Chapter 2: Survey of the Sum of Lognormal . . . . . . . . . . 62.1 The Sum of Lognormal RVs . . . . . . . . . . . . . . . . . . . 62.2 Approximating Using a Single Lognormal RV . . . . . . . . . 72.2.1 Wilkinson’s Method . . . . . . . . . . . . . . . . . . . 72.2.2 Schwartz and Yeh’s Method . . . . . . . . . . . . . . . 82.2.3 Minimax Approximation Method . . . . . . . . . . . . 102.2.4 MGF Matching Method . . . . . . . . . . . . . . . . . 132.2.5 Cumulants Matching Method . . . . . . . . . . . . . . 142.3 Approximating Using a Single Non-Lognormal RV . . . . . . 16viTABLE OF CONTENTS2.3.1 LSQ Method . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Curve Fitting Method . . . . . . . . . . . . . . . . . . 182.3.3 LSN Approximation . . . . . . . . . . . . . . . . . . . 202.3.4 Type-IV Pearson Method . . . . . . . . . . . . . . . . 232.3.5 LSG Approximation Method . . . . . . . . . . . . . . 242.4 Bounds of the Sum of Lognormal RVs . . . . . . . . . . . . . 282.4.1 Farley’s Upper Bound . . . . . . . . . . . . . . . . . . 282.4.2 Slimane’s Upper and Lower Bounds . . . . . . . . . . 292.4.3 Tellambura’s Upper and Lower Bounds . . . . . . . . 312.4.4 Xiao’s Upper and Lower Bounds . . . . . . . . . . . . 332.5 Asymptotic CDF of Lognormal Sum . . . . . . . . . . . . . . 352.5.1 Best Lognormal Fit Method . . . . . . . . . . . . . . . 352.5.2 Zhu’s Right Tail Approximation in Asymptotic Scenario 362.5.3 Asymptotic CDF Based on Minimum Distance Theoryand Marcum Q-Function . . . . . . . . . . . . . . . . . 382.6 Comparisons of Methods . . . . . . . . . . . . . . . . . . . . . 45Chapter 3: Application of the Sum of lognormal RVs . . . . . 493.1 Co-Channel Interference Analysis . . . . . . . . . . . . . . . . 493.2 Diversity Combining . . . . . . . . . . . . . . . . . . . . . . . 503.2.1 MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2.2 EGC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Chapter 4: The Optimal Marcum Q-function for the Approx-imation of the Sum of Two Lognormal RVs . . . . 544.1 Sum of Two Independent Lognormal RVs . . . . . . . . . . . 544.2 Sum of Two Correlated Lognormal RVs . . . . . . . . . . . . 624.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 695.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71viiList of TablesTable 2.1 Sum Distribution Cumulants (y = eσ2) [Sch77]. . . . . 16Table 2.2 Reference Lognormal Distribution LR(xr, σr) from Cu-mulant Matching [Sch77]. . . . . . . . . . . . . . . . . 16Table 2.3 The Coefficient for the Approximation Φ(α0−α1e−α2γ)[BR04] 19Table 2.4 Using a Distribution to Approximate Lognormal SumDistribution. . . . . . . . . . . . . . . . . . . . . . . . 47Table 2.5 Different Bounds . . . . . . . . . . . . . . . . . . . . . 48Table 2.6 Different Asymptotic Methods . . . . . . . . . . . . . 48viiiList of FiguresFigure 3.1 Frequency reuses of cellular radio systems . . . . . . . 49Figure 3.2 Concept of diversity combining (gn is the weight foreach branch) . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.1 The boundary of integration region in the X1 − X2plane, for s′ = 0.25, s′ = 1, s′ = 5. . . . . . . . . . . . 57Figure 4.2 Using a circular region to approximate exponentialregion . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 4.3 Exponential region cannot cover the circular region ifr >√2. . . . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.4 The difference between exponential and circular re-gion when r <√2 . . . . . . . . . . . . . . . . . . . . 62Figure 4.5 The probability distribution left tail of the sum of twoindependent lognormal RVs using simulation and twoMarcum Q-methods . . . . . . . . . . . . . . . . . . . 63Figure 4.6 The curve of the relationship between a and ρ inEquation (4.30c) . . . . . . . . . . . . . . . . . . . . 64Figure 4.7 Γexp,cor with different a and s′cor . . . . . . . . . . . . 65Figure 4.8 Exponential boundary Γexp,cor,0 and the correspond-ing optimal circular boundary Γcir,cor,0 when a takesdifferent values . . . . . . . . . . . . . . . . . . . . . 67ixList of AcronymsCDF Cumulative Distribution Function. 3EGC Equal-Gain Combining. 2FSO Free Space Optical. 1IID Independent and Identically Distributed. 20LSG Log Shifted Gamma. 4LSN Log Skew Normal. 4LSQ Least Squares Quadratic. 4MGF Moment Generating Function. 4MRC Maximum-Ratio Combining. 2PDF Probability Density Function. 3RF Radio Frequency. 1RV Random Variable. 1SC Selection Combining. 2SIR Signal-to-Interference Ratio. 1SN Skew Normal. 20SNR Signal-to-Noise Ratio. 50xAcknowledgementsI am deeply grateful to my supervisor Dr. Julian Cheng for his constantguidance, encouragement and friendship.I owe particular thanks to Dr. Tao Shan and Dr. Nan He for theirvaluable suggestions and selfless help.Finally, I want to thank my family for their love and support.xiTo my lovexiiChapter 1Introduction1.1 Background and MotivationLognormal distribution is widely used to describe the channel charac-teristics in Radio Frequency (RF) communication system and Free SpaceOptical (FSO) communication system. When the outputs of different chan-nels need to be combined, the sum of lognormal Random Variable (RV)soften describes the statistical behavior of combiner output.The first application of lognormal sum is for interference analysis inthe frequency reuses of cellular radio systems. The frequency spectrumis a valuable resource, which can be divided into non-overlapping spectrumbands assigned to different cells. With a certain geographical distance, thesefrequency bands are reused. The method of specifying frequency rangesis used multiple times in the same radio system, thereby significantly in-creasing system capacity and frequency utilization without increasing theallocated bandwidth. However, frequency-reuse will introduce co-channelinterference from other transmitters using the same frequencies. The sys-tem performance will be limited by the Signal-to-Interference Ratio (SIR).The co-channel interference can be reduced to a tolerance level by separat-ing the two same-frequencies transmitters significantly far away. But sep-arating transmitters often leads to reduced capacity. Thus, quantitativelydescribing the interference characteristic helps balance the system capacityand co-channel interference. Thus, we can calculate the optimal transmitterspacing and frequency reuse scheme. Since characteristics per channel canbe often described by a lognormal distribution, total co-channel interferenceis the sum of multiple lognormal RVs [Rap96].The second application of lognormal sum is for performance analysis ofdiversity reception in RF and FSO communication systems. In RF commu-nication systems, the lognormal distribution is often used to describe chan-nel large-scale fading (shadowing) and, recently small-scale fading in indoorultra-wideband communications [HCHT09, Sha17, ZKA03]. In FSO commu-nication systems, optical signals suffer from intensity fluctuations caused byatmospheric turbulence. The lognormal distribution is the most commonly11.2. LOGNORMAL RV AND OVERVIEW OF LOGNORMAL SUM LITERATUREused for the weak turbulence scenario [ZWK22, DSS17]. To counteract thefluctuations introduced by fading or turbulence, we often use diversity recep-tion to improve the signal-to-noise ratio, where outputs of different branchesare combined into a single output by some diversity combining algorithms:Selection Combining (SC), Maximum-Ratio Combining (MRC), and Equal-Gain Combining (EGC). In the lognormal fading channel, MRC and EGCboth require the characteristics of the sum of lognormal RVs. Knowing thelognormal-sum distribution helps analyze the output of the combiner, whichis critical to analyze the system performance, such as outage probability,channel capacity, and bit-error rate [LSA09, TSC17, AS02, SSPK11].How to derive the distribution of a sum of lognormal RVs is a longstand-ing problem because the distribution is not known to have a closed-form ana-lytical expression. Therefore, many popular methods for approximating thesum of lognormal RVs have been proposed, and these methods include theFenton-Wilkinson method, Schwartz-Yeh method, Farley’s methods, mini-max method, Marcum Q-function and so on, each having its correspondingshortcomings.1.2 Lognormal RV and Overview of LognormalSum LiteratureIn this section, we review some fundamental concepts related to thelognormal RV and brifely review the sum of lognormal literature. If X isa lognormal distributed RV, Y = lnX is a Gaussian RV. Equivalently, ifY is Gaussian distributed, then X = eY is a lognormal distributed RV.This definition implies that a lognormal distributed RV takes positive realvalues only. The lognormal distribution naturally arises from many science,engineering, and finance disciplines.Let Z be a standard Gaussian RV and let µ and σ > 0 be two realnumbers. A new RV X that followsX = eµ+σ·Z (1.1)can be called a lognormal RV having parameters µ and σ regardless of thebase of the logarithmic or exponential function. The mean µX and variance21.2. LOGNORMAL RV AND OVERVIEW OF LOGNORMAL SUM LITERATUREσ2X of the lognormal RV X followsµ = ln µ2X√µ2X + σ2X (1.2a)σ = ln(1 +σ2Xµ2X). (1.2b)Thus, X ∼ Lognormal(µX, σ2X). If the natural logarithm of X is Gaussiandistributed with mean µ and variance σ2, i.e., ln (X) ∼ N (µ, σ2). Let Φand ϕ be the Cumulative Distribution Function (CDF) and the ProbabilityDensity Function (PDF) of the standard Gaussian RV, respectively, then wehave PDF of lognormal RV X asfX(x) =ddxPr{X ≤ x} = ddxPr{lnX ≤ lnx}=ddxΦ(lnx− µσ)= ϕ(lnx− µσ)ddx(lnx− µσ)= ϕ(lnx− µσ)1x · σ=1xσ√2πexp(−(lnx− µ)22σ2).(1.3)Furthermore, the CDF isFX(x) = Φ((lnx)− µσ)(1.4a)=12[1 + erf(lnx− µ√2σ)](1.4b)=12erfc(− lnx− µ√2σ)(1.4c)where erf(·) and erfc(·) are the error function and the complementary er-ror function, respectively. If we consider a Gaussian random vector X =[X1, X2, · · · , XN ]T having mean vector µ and covariance matrix Σ, Y =[Y1, Y2, · · · , YN ]T = [exp (X1), exp (X2), · · · , exp (XN )]T is a lognormal dis-tributed random vector, where the mean vector and covariance matrix can31.2. LOGNORMAL RV AND OVERVIEW OF LOGNORMAL SUM LITERATUREbe calculated byE[Y ]i = eµi+12Σii (1.5a)V ar[Y ]ij = eµi+µj+12(Σii+Σjj)(eΣij − 1) (1.5b)where Σii is the (i, i)th entry of the covariance matrix Σ, E[Y ]i is the ex-pectation of Yi, and V ar[Y ]ij is the (i, j)th entry of the covariance matrixV ar[Y ]. Furthermore, all moments of the lognormal distribution exists andcan be given asE[Xn] = enµ+n2σ2/2. (1.6)However, unlike the standard Gaussian distribution, the distribution ofa lognormal RV cannot be determined by its moments. The defined MomentGenerating Function (MGF) doesn’t exist in a neighborhood of zero [Bil12].Moreover, a closed-form characteristic function of a lognormal is unknownsince its Taylor series diverges:∞∑n=0(it)nn!enµ+n2σ2/2 (1.7)where i2 = −1.Since the distribution of the sum of lognormal RVs is not lognormaldistributed and does not have a closed-form expression, various approxima-tion methods and bounding techniques have been developed. Based on thebasic statistic knowledge of lognormal distribution, we investigate relatedresearch papers and compare the approximation methods for the sum of thelognormal distribution.The previously proposed methods can be broadly classified into fourcategories. First, based on the idea of approximating the distribution ofa sum of independent lognormal RVs by a distribution of another lognor-mal RV, researchers have applied various approaches to determine the pa-rameters of the approximate lognormal distribution. Wilkinson [Fen60],Schwartz-Yeh [SY82], Farley [BADM94], and Beaulieu-Xie [BX04] proposedvarious distribution approximations based on this approach. MGF match-ing and cumulant matching are also used in approximation. Second, theLog Skew Normal (LSN) method [LWCW11], Type-IV Pearson method[ZS06], Least Squares Quadratic (LSQ) method [ZD06], Log Shifted Gamma(LSG) [LLN07] method and curve fitting method [BR04] approximate thelognormal sum using various non-lognormal RVs. Third, various boundingtechniques [ZZW+20, ZCW+18] have been developed to upper-bound and41.3. THESIS ORGANIZATION AND CONTRIBUTIONSlower-bound the distribution of a lognormal sum. Finally, since the tail dis-tribution of the lognormal sum is challenging to describe, several closed-formasymptotic methods were proposed to approximate the distribution near theorigin. Moreover, we construct a geometric model to analyze the lognormalsum, and this approximation is accurate over a wide variable range.1.3 Thesis Organization and ContributionsThis thesis contains five chapters. The summary of each chapter and itscontributions are presented as follows.As the shortage of frequency resources, frequency reuse becomes a promis-ing alternative technology for telecommunication applications. However, thereceiver suffers from severe co-interference introduced by other radio trans-mitters using the same frequency channel. To overcome these limitations andimprove the system performance, it is important to study the characteriza-tions of lognormal fading, the sum of lognormal, as well as approximation ofthe sum of lognormal. Chapter 1 introduces several background knowledgeand recent development on the sum of the lognormal distribution.Chapter 2 focuses on the approximation methods of the sum of the log-normal distribution. First, we introduce several approaches of approximateusing a single RV. Second, we present bounds and asymptotic methods of thesum of lognormal distribution modelling. Detailed approximation methodsand representative research works have been demonstrated in each method.In Chapter 3, we investigate the applications of the sum of the the dis-tribution of a sum of lognormal RVs. Since most wireless communicationsystems adopt diversity mechanisms, several receiving models mentionedin this section, such as MRC and EGC, have been widely implemented inpractical applications. We will briefly introduce the characterizations ofeach scheme and demonstrate their applications.In Chapter 4, we propose a geometric method to analytically calculatethe maximum circular region for approximating the CDF of the sum of twoIID lognormal RVs via the Marcum Q-function. The result improves theaccuracy of the Marcum Q approximation based on numerical optimization.Then, we extend the proposed geometric method for a sum of two correlatedlognormal RVs. The result coincides with the results obtained by the third-order Taylor series expansion, i.e., curvature matching.Chapter 5 summarizes the entire thesis and states our contributions tothis work. In addition, future work and potential research directions of thesum of lognormal distribution are also suggested.5Chapter 2Survey of the Sum ofLognormalIn this chapter, we will present the basic knowledge related to the sumof lognormal RVs. Then, we survey various methods to approximate thedistribution of sum of lognormal RVs, and these methods include approx-imation using a single lognormal RV, approximation using a different RV,boundaries of the lognormal sum, and asymptotic analysis.2.1 The Sum of Lognormal RVsAccording to probability theory, if the logarithm of a continuous RV isGaussian distributed, the RV is lognormal distributed. Let L1, L2, · · · , LNbe N independent, but not necessarily identical distributed lognormal RVsand let Xi = 10 log10 Li. Then Xi is a Gaussian RV with a PDFfXi(x) =1√2πσxiexp(−(x−mxi)22σ2xi)(2.1)where µxi and σxi are the mean and standard deviation of Xi, it is moreconvenient to use the natural logarithm of Li. Let Yi = lnLi, thenfYi(y) =1√2πσyiexp(−(y − µyi)22σ2yi)(2.2)where µyi and σyi are the mean and standard deviation of Gaussian RV Yi.The relationship between Xi and Yi follows that Yi = λXi, µyi = λµxi , andσyi = λσxi , where λ = 0.1 ln(10) = 0.23026. The PDF of Li isfLi(u) =1√2πuσLiexp(− ln2(u/µLi)2σ2Li)(2.3)where lnµLi and σLi are the mean and standard deviation of the correspond-ing Gaussian distribution. A lognormal RV can well approximate the sum62.2. APPROXIMATING USING A SINGLE LOGNORMAL RVof lognormal RVs [SS96], i.e.,L = L1 + L2 + · · ·+ LN = eY1 + eY2 + · · ·+ eYN ≈ eZ (2.4)where Z is another Gaussian distributed RV.Generally, the distribution of the sum of lognormal does not have closed-form expressions. However, various researchers proposed different approxi-mation methods finding the distribution of a sum RVs. We will introducethese methods in the following section.2.2 Approximating Using a Single Lognormal RVThe widely accepted assumption is that a single lognormal RV can de-scribe the statistical behaviors of a lognormal sum well. Based on thisassumption, various approximate methods have been developed to calculatethe CDF or the moments of a sum of independent lognormal RVs, includingWilkinson’s, Schwartz-Yeh’s, the minimax, MGF matching, and cumulantsmatching method. These methods are introduced briefly in this section.2.2.1 Wilkinson’s MethodUsing the Wilkinson’s method [Fen60], the mean and standard deviationof RV Z can be calculated by matching the first and second moments of bothsides of Equation (2.4)E[eZ ] = E[L1 + L2 + · · ·+ LN ], (2.5a)E[e2Z ] = E[(L1 + L2 + · · ·+ LN )2]. (2.5b)Since the moments of a lognormal RV can be calculated using the MGF:E[Ln] = E[(eY )n] = enµY +12n2σ2Y . (2.6)Combining Equation (2.6) with Equation (2.5a) and (2.5b), we can obtaineµZ+12σ2Z = E[L1] + E[L2] + · · ·+ E[LN ]=N∑i=1eµYi+12σ2Yi≜ u1(2.7)72.2. APPROXIMATING USING A SINGLE LOGNORMAL RVande2µZ+2σ2Z =N∑i=1E[L2i ] + 2N−1∑i=1N∑j=i+1E[Li]E[Lj ]=N∑i=1σ2Li + u21≜ u2 + u21(2.8)whereσ2Li = E[L2i ]− E[Li]2 (2.9a)u2 =N∑i=1σ2Li =N∑i=1e2µYi+σ2Yi (eσ2Yi − 1). (2.9b)The mean and standard deviation can be solved by computing Equations(2.7) and (2.8) and obtained asµZ = lnu1 − σ2Z/2 (2.10a)σ2Z = ln(u2u1+ 1). (2.10b)Wilkinson’s method gives accurate approximation in some practical prob-lems. However, this approach is valid for limited range of small values of σonly. The Wilkinson approach may break down when σ > 4 dB.2.2.2 Schwartz and Yeh’s MethodSchwartz and Yeh’s method also uses the first two moments [SY82]. Thismethod calculates the exact expressions for the first two moments and gives ahigher approximation accuracy than the much simpler Wilkinson’s approach.Since a logormal RV can well approximate the sum of two lognormal RVs[Fen60], we can calculate the moments of the sum of multiple lognormal RVs(N > 2) by using a recursive procedure. The calculation procedure is shownaseZ = eY1 + eY2 + · · ·+ eYN= eZ2 + eY3 + · · ·+ eYN= eZN−1 + eYN(2.11)where eZ is a lognormal RV, and Y1, Y2, . . . , YN are Gaussian RVs. First,we need to calculate the lognormal sum for a bivariate case and then adopt82.2. APPROXIMATING USING A SINGLE LOGNORMAL RVa recursive procedure to calculate the result of the multiple lognormal case.Using one lognormal RV eZ to describe the sum of two lognormal, we canobtaineZ = eY1 + eY2 (2.12)where Y1 and Y2 are Gaussian RVs having means µY1 , µY2 and standarddeviations σY1 , σY2 , respectively. We can define Yd = Y2 − Y1 , where Yd isa new Gaussian RV so thatµYd = µY2 − µY1 (2.13a)σ2Yd = σ2Y1 + σ2Y2 . (2.13b)After some complicated calculation [SY82], the mean value of Z is given byµZ = µY1 +G1 (2.14)whereG1 = µYdΦ(µYdσYd)+σYd√2πe−µ2Yd/(2σ2Yd)+∞∑k=1ckek2σ2Yd/2[ekµYdΦ(−µYd − kσ2YdσYd)+ T1] (2.15)withT1 = e−kµYdΦ(µYd − kσ2YdσYd)(2.16a)ck =(−1)k+1k. (2.16b)The variance can be given byσ2Z = σ2Y1 −G21 − 2σ2Y1G3 +G2 (2.17)whereG2 =∞∑k=1bkT2 +[1− Φ(−µYdσYd)](µ2Yd + σ2Yd) +µYdσYd√2πe−µ2Yd/(2σ2Yd)+∞∑k=1bke−(k+1)µYd+(k+1)2σ2Yd/2Φ(µYd − σ2Yd(k + 1)σYd)− 2∞∑k=1cke−µYdk+k2σ2Yd/2[µYdΦ(−µYkσYd)− σYd√2πe−µ2Yk/(2σ2Yd)](2.18)92.2. APPROXIMATING USING A SINGLE LOGNORMAL RVG3 =∞∑k=0(−1)kek2σ2Yd/2T1 +∞∑k=0(−1)kT2 (2.19)withT2 = eµYd (k+1)+(k+1)2σ2Yd/2Φ(−µYd − (k + 1)σ2YdσYd)(2.20a)bk =2(−1)k+1k + 1k∑j=11j(2.20b)µYk = −µYd + kσ2Yd . (2.20c)The Schwartz and Yeh’s method is computationally intensive, but thismethod provides more accurate approximation than theWilkinson’s method.This iterative method was subsequently extended to the case of correlatedlognormal components [Saf93].2.2.3 Minimax Approximation MethodBeaulieu and Xie proposed a new approximation method [BX04], whichuses one lognormal RV to approximate a broader range of CDF of lognormalsums than the previous works. Inspired by the fact that a Gaussian CDFcan be plotted as a straight line on a ’probability paper’ [Dev15], Beaulieuand Xie defined a lognormal probability paper, where a lognormal CDF plotsas a straight line. Thus, on lognormal probability paper, approximating alognormal sum distribution with single lognormal distribution is equivalentto approximating a curve of lognormal sum CDF with a straight line oflognormal CDF. This approach uses a straight line that minimizes the max-imum deviation from the numerical curve of lognormal sums. Accordingto an approximation theory [Che66], we can find a unique solution for thecontinuous function defined on a specific finite interval. Moreover, the al-ternation theorem [Jan70] provides the mathematic theory for the minimaxapproximation.Alternation Theorem: Let f(x) belong to C[a, b] and p(x) be a poly-nomial of degree n. Let En = maxa≤x≤b |f(x)−p(x)| and ε(x) = f(x)−p(x).A necessary and sufficient condition that p(x) be the unique best uniformapproximation to f(x) is that there are at least n+ 2 points a ≤ x1 < x2 <· · · < xn+2 ≤ b where ε(x) assumes the values ±En and with alternating102.2. APPROXIMATING USING A SINGLE LOGNORMAL RVsigns:ε(xi) = ±En, i = 1, 2, · · · , n+ 2 (2.21a)ε(xi) = −ε(xi+1), i = 1, 2, · · · , n+ 1. (2.21b)Let F (x) be the CDF of an arbitrary RV and Φ−1(x) be the inversefunction of the standard Gaussian CDF, then F (x) can be transformed tog(x) = Φ−1(F (x)). (2.22)For a Gaussian RV X with N (µ, σ2), its CDF FX(x) can be transformedby Equation (2.22) asg1(x) = Φ−1 (FX(x)) =xσ− µσ(2.23)where x and g1(x) have a linear relationship.For the corresponding lognormal RV eX , the same transformation of itsCDF FL(x) leads tog2(x) = Φ−1(FL(x)) =1σlnx− µσ(2.24)where lnx and g2(x) have a linear relationship. Thus, the mapping of theCDF F (x) of an arbitrary RV from original probability paper to lognormalprobability paper can be defined as{x, F (x)} 7→ {ln (x), g(x)} = {ln (x),Φ−1(F (x))} . (2.25)The lognormal CDF on lognormal probability paper is a straight line.Based on the mapping above, approximating the lognormal sum distribu-tion using a lognormal distribution on original probability paper means look-ing for a specific straight line to ’match’ the curve of lognormal sum distribu-tion on lognormal probability paper. Let {FS(xi), i = 1, 2, ...,m} be the dis-tinct values of the CDF of lognormal sum, which can be obtained by simula-tion or numerical integration. After mapped to lognormal probability paper,{xi, FS(xi)} leads to {ln (xi), gS(xi)} = {ln (xi),Φ−1(FS(xi))}. A straightline on the lognormal probability paper is defined as p(x) = c0+c1 lnx. The’match’ criterion is minimizing the maximum value of |gS(xi)− p(xi)|, i.e,minc0,c1max1≤i≤m|gS(xi)− (c0 + c1 lnxi)|, 0 < a = x1 < x2 < · · · < xm = b <∞.(2.26)112.2. APPROXIMATING USING A SINGLE LOGNORMAL RVBased on the Alternation Theorem, at least three points between a andb have the maximum deviation E1 > 0, i.e.,gS(xm,k)− p(xm,k) = ±E1, k = 1, 2, 3 (2.27)where xm,1 < xm,2 < xm,3. Meanwhile, gS(x) is a concave function withdg2S(x)/d(lnx)2 < 0 [BX04], which means dgS(x)/d(lnx) is a monotonicfunction. Thus, we havegS(xm,1)− p(xm,1) = −E1gS(xm,2)− p(xm,2) = +E1gS(xm,3)− p(xm,3) = −E1.(2.28)We can first find the relationship between xm,2 and the to-be-determinedparameter c1. After differentiating both sides of Equation (2.27), we canobtaindgSd(lnx)(xm,k)− dpd(lnx)(xm,k) =dgSd(lnx)(xm,k)− c1 = 0 (2.29)which has only one root on [a, b] based on the Alternation Theorem. Thus,dgSd(lnx)(xm,2) = c1. (2.30)The other two large deviation points can be found at the end of theinterval, that is xm,1 = a, xm,3 = b. After some computation [Che66], wecan find the coefficients c1 and c0c1 =gS(b)− gS(a)b− a (2.31a)c0 =12[gS(a) + gS(xm,2)]− a+ ln(xm,2)2· gS(b)− gS(a)b− a (2.31b)where xm,2 can be obtained by combining Equations (2.30) and (2.31a), i.e.,dgSd(lnx)(xm,2) =g(b)− g(a)b− a . (2.32)After connecting (ln (xm,1), g(xm,1)) and (ln (xm,3), g(xm,3)) on the lognor-mal probability paper, we can form a straight line AB with the slope c1.Another point, C, on the curve can also be found to keep the tangent ofthe curve on C parallel to AB. Thus, the required straight line p(x) is the122.2. APPROXIMATING USING A SINGLE LOGNORMAL RVline in the middle between line AB and point C. The mean µ∗ and standarddeviation σ∗ of the required lognormal RV can be calculated byµ∗ = −c0/c1 (2.33a)σ∗ = 1/c1. (2.33b)The approximations obtained by previous methods are invalid over a widerange of parameters and the approximations given by minimax are good for6-12 dB spreads. However, the maximum errors increase when the numberof lognormal RVs increases.2.2.4 MGF Matching MethodFrom the previous Wilkinson’s method, the mean and standard deviationof RV Z can be calculated by matching the first and second moment. As forthe MGF, if all the individual RVs are independent, then the PDF of thelinear combination of these RVs is the convolution of the PDF of each RVs.The MGF of a RV Y is defined asΨY (s) =∫ ∞0exp(−sy)pY (y)dy (2.34)where {Yk}Kk=1 are independently distributed, and the MGF isΨ∑Kk=1 Yk(s) =K∏k=1ΨYk(s). (2.35)Since the lognormal MGF has no closed-form expression, by using the Gauss-Hermite integration method, the MGF of a lognormal RV Y for real s canbe written asΨY (s) =∫ ∞0exp(−sy) ξyσx√2πexp[−(ξ loge y − µx)22σ2x]dy=∫ ∞−∞1√πexp[−s exp(√2σz + µξ)]exp(−z2)dz=N∑n=1ωn√πexp[−s exp(√2σxan + µxξ)]+RN≈N∑n=1ωn√πexp[−s exp(√2σxan + µxξ)]≜ ΨˆY (s)(2.36)132.2. APPROXIMATING USING A SINGLE LOGNORMAL RVwhere µx is the mean and σx is the standard deviation, RN is the remainderterm [MWMZ07, MMWZ06]. By matching the MGF of Y with the MGFof the sum of lognormal at the s1 and s2, one obtainsN∑n=1ωn√πexp[−si exp(√2σxan + µxξ)]=K∏k=1ΨˆYk(si;µxk , σxk), for i = 1 and 2.(2.37)We can then obtain the values of µx and σx by solving Equation (2.37).The accuracy will increase when integration order N goes up; whereas smallN decreases computation complexity. The MGF method can accurately ap-proximate both the CDF and the complementary CDF of the lognormal sumdistribution over a wide range of parameters. However, we need to choosethe value of s1 and s2 properly to achieve high approximation accuracy.2.2.5 Cumulants Matching MethodSchleher proposed a new approximation method to calculate the distribu-tion of the sum of independent lognormal RVs based on cumulant matching[Sch77]. For an arbitary RV Z with PDF pZ(z), the characteristic functionof pZ(z) is given by [Cra46]ϕZ(t) = exp( ∞∑n=1χZ,nn!(jt)n)(2.38)where χZ,n is the n-th cumulant of pZ(z) and can be calculated using themoments about the origin. The cumulant of the sum of N independent,identically distributed variables is simply N times the cumulant of basicdistribution. Furthermore, the difference between the cumulants of two RVsenables us to represent one random by the other:FS(z) = FR(z) +∞∑i=1(−1)iEii!f(i−1)R (z)≈ FR(z) +6∑i=1(−1)iEii!f(i−1)R (z)(2.39)where FS(z) is the distribution of the target RV S. The FR(z) and fR(z) areCDF and PDF of the reference RV R, respectively. The f(i−1)R is the (i−1)-th order derivation of fR(z), and Ei are the error coefficient given in terms142.2. APPROXIMATING USING A SINGLE LOGNORMAL RVof the difference between the cumulants of S and R (χϵ,n = χS,n − χR,n).Expressions for the first six error coefficients areE1 = χϵ,1E2 = χϵ,2 + χ2ϵ,1E3 = χϵ,3 + 3χϵ,1χϵ,2 + χ3ϵ,1E4 = χϵ,4 + 4χϵ,3χϵ,1 + 6χϵ,2χ2ϵ,1 + 3χ2ϵ,2 + χ4ϵ,1E5 = χϵ,5 + 5χϵ,4χϵ,1 + 10χϵ,3χϵ,2 + 10χϵ,3χ2ϵ,1 + 15χ2ϵ,2χϵ,1+ 10χϵ,2χ3ϵ,1 + χ5ϵ,1E6 = χϵ,6 + 6χϵ,5χϵ,1 + 15χϵ,4χϵ,2 + 15χϵ,4χ2ϵ,1 + 10χ2ϵ,3 + 60χϵ,3χϵ,2χϵ,1+ 20χϵ,3χ3ϵ,1 + 15χ3ϵ,2 + 45χ2ϵ,2χ2ϵ,1 + 15χϵ,2χ4ϵ,1 + χ6ϵ,1.(2.40)In the sum of lognormal problem, we need to find the CDF ofS =N∑n=1Xn (2.41)where {Xn(xm, σ), n = 1, · · · , N} are independent and identically distributedlognormal RVs with PDFfX(x) =1√2πσxexp(− ln2 xxm2σ2). (2.42)Thus, in this IID case, the cumulant of S can be easily calculated using themoments of Xn, as shown in Table 2.1. We select a lognormal distributionLR(xr, σr) as the reference distribution in Equation (2.39). The lognormalparameter (xr, σr) can be obtained by matching arbitrary two cumulantsof reference distribution to the corresponding cumulant of lognormal sumdistribution in Table 2.1. If we match successively the first and second; sec-ond and third; and third and fourth cumulants of the reference distributionagainst the cumulants of the sum distribution. We can obtain three differentreference distributions, i.e., {LR(xr,i, σr,i)|i = 1, 2, 3}, as shown in Table 2.2.Each reference distribution can be used in Equation (2.39) to approximatethe sum distribution for a specific range of z. The boundary of z, where thereference distribution LR(xr,i, σr,i) is shifted to the next LR(xr,i+1, σr,i+1) isgiven byz =[xσr,i+1r,ixσr,ir,i+1] 1σr,i+1−σr,i. (2.43)152.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVCumulant matching uses different reference lognormal distributions andcorresponding error correction terms to approximate different parts of thelognormal distribution.Beaulieu et al. proved that the reference lognormal CDF perfectly rep-resents the sum of lognormal without the correction terms [BADM94]. Thissimplified approach also gives excellent results especially for small standarddeviation case.Table 2.1: Sum Distribution Cumulants (y = eσ2) [Sch77].Cumulant FormulaχS,1 nxmy12χS,2 nx2my(y − 1)χS,3 nx3my32 (y3 − 3y + 2)χS,4 nx4my2(y6 − 4y4 − 3y3 + 12y − 6)Table 2.2: Reference Lognormal Distribution LR(xr, σr) from CumulantMatching [Sch77].CumulantMatchingVariable σ2r xr1-2 d =χS,2χ2S,2ln (d+ 1) χS,1 exp (−σ2r2 )2-3 k = 4 +χ2S,3χ3S,2ln (s1 + s2 − 1) 2√χS,2e2σ2r−eσ2rs1,2 ={k2(1± (1− 4k) 12)− 1} 133-4 a = χ34S,4/χS,3Fig.4 (σr, F )[Sch77]3√χ3SFFig.4 (σr, F ) [Sch77]2.3 Approximating Using a SingleNon-Lognormal RV2.3.1 LSQ MethodIn Section 2.2.3, Beaulieu and Xie tried to use a lognormal RV, whoseCDF is a straight line on the lognormal probability paper, to approximate162.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVthe lognormal sum, whose CDF is a curve on the lognormal probabilitypaper. To improve the accuracy of the curve approximation of lognormalsum, Zhao and Ding used a polynomial curve and the LSQ approximation[ZD06].For a lognormal sum RV, Z, the corresponding dB value of Z, definedas Γ, isΓ = 10 log10 Z =lnZβ(2.44)where β = ln 10/10 ≈ 0.23026. The relationship between the distributionfunctions of Z and Γ isFZ(z) = P (Z ≤ z) = P (eβΓ≤z)= P(Γ ≤ ln zβ)= FΓ(ln zβ).(2.45)The CDF of Γ plotted on a lognormal probability paper can be obtained bythe following transformation:G(γ) ≡ Φ−1(FΓ(γ)) = Φ−1(FZ(eβγ)) = gZ(eβγ) (2.46)where Φ(x) is the distribution function of the standard Gaussian RV andgZ(z) = Φ−1(FZ(z)).Since the least squares approximation based on 2-norm is defined as thefollowing optimization problem:minp∈Pn∥G− p∥2 = minp∈Pn(∫ ba[G(γ)− p(γ)]2dγ) 12(2.47)where Pn is the space of all polynomials of degree at most n. Thus, the leastsquares approximation by quadratic functions isminc0,c1,c2r(c0, c1, c2) = minc0,c1,c2∫ ba[G(γ)− c0 − c1γ − c2γ2]2dγ (2.48)where a and b are the required interval of interest.By taking the derivative of γ with respect to {c0, c1, c2}, we can obtainthe coefficients shown asc0 =3(b− a)5 {3[(a+ b)4 + 4a2b2]∫ baG(γ)dγ− 12(a+ b)(a2 + 3ab+ b2)∫ baG(γ)γdγ+ 10(a2 + 4ab+ b2)∫ baG(γ)γ2dγ}(2.49)172.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVc1 =12(b− a)5 {4[4(a+ b)2 − ab]∫ baG(γ)γdγ− 3(a+ b)[(a2 + 3ab+ b2)∫ baG(γ)dγ+ 5∫ baG(γ)γ2dγ]}(2.50)c2 =30(b− a)5 {(a2 + 4ab+ b2)∫ baG(γ)dγ− 6(a+ b)∫ baG(γ)γdγ + 6∫ baG(γ)γ2dγ}.(2.51)From the optimal quadratic polynomial p(γ) = c0 + c1γ + c2γ2, the CDF ofthe sum of lognormal γ can be obtained asF˜Γ(γ) = Φ (p(γ)) = Φ(c0 + c1γ + c2γ2). (2.52)The corresponding PDF isfˆΓ(γ) =c1 + 2c2γ√2πexp[−(c0 + c1γ + c2γ2)22]U(γ) (2.53)where U(γ) is the unit step function.2.3.2 Curve Fitting MethodSimilar to Sections 2.2.3 and 2.3.1, the approximation approach in thissection is also based on matching a specific curve to the curve of a lognormalsum [BR04]. Different from the straight line and quadratic curve used before,the approximating function used here is an exponential function, i.e.,f(γ) = α0 − α1e−α2γ (2.54)where γ is in dB; α0, α1, andα2 are constant parameters to be determinedby a nonlinear least squares method. The optimal values are given in theTable 2.3.The CDF of lognormal sum Z isFZ(z) = Pr{Z ≤ z} = Φ(f(ln zλ))= Φ(α0 − α1z−α2λ) (2.55)182.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVTable 2.3: The Coefficient for the Approximation Φ(α0 − α1e−α2γ)[BR04]Nσ = 12 dB σ = 6dBα0 α1 α2 α0 α1 α22 23.91 24.67 0.00430 25.64 26.60 0.008283 16.33 17.67 0.00709 15.95 17.74 0.01554 12.69 14.51 0.00995 13.66 16.16 0.01995 11.22 13.48 0.0120 11.08 14.37 0.02706 10.13 12.81 0.0140 10.76 14.69 0.02947 9.46 12.53 0.0157 10.04 14.67 0.03348 8.94 12.38 0.0172 9.47 14.83 0.03729 8.52 12.33 0.0187 9.09 15.17 0.040510 8.21 12.37 0.0200 8.75 15.58 0.043811 8.03 12.53 0.0210 8.44 16.05 0.047212 7.77 12.62 0.0223 8.16 16.61 0.050613 7.62 12.79 0.0232 8.02 17.24 0.053014 7.42 12.94 0.0243 7.90 17.91 0.055315 7.29 13.12 0.0252 7.65 18.67 0.059016 7.14 13.31 0.0262 7.65 19.34 0.060017 7.03 13.52 0.0271 7.42 20.26 0.063818 6.94 13.73 0.0279 7.31 21.13 0.066319 6.83 13.95 0.0288 7.20 22.07 0.068820 6.75 14.18 0.0295 7.22 22.80 0.0696where λ = ln 10/10. Taking the derivative, we have the PDF of the sum as:fZ(z) =α2α2z−(α2/λ+1)λ√2πexp(−12(α0 − α1z−α2/λ)2)(2.56)fZ(γ) =α1α2√2πexp(−12(α0 − α1e−γα2)2 − γα2). (2.57)The Equation (2.57) was corrected by Li [ZD06]. Furthermore, althoughthis method provides accurate closed form approximations of the sum oflognormal distribution, the parameters are unknown until the PDF andCDF of the sum of lognormal distribution have been obtained.In essence, Equation (2.55) can be also considered as the generalizedlognormal distribution [LAWM07]. To further improve the accuracy, Liuet al. approximated the lognormal sum by using two-component mixture192.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVlognormal RV, whose CDF isFm(x, θ) = γF (x, µ1, σ1) + (1− γ)F (x, µ2, σ2) (2.58)where θ = (µ1, σ1, µ2, σ2) and Fm(x, µi, σi) = Φ(ln (x)−µiσi)is the CDF ofthe i-th component.To approximate the CDF of lognormal sum FL(l), Liu proposed andsolved the following nonlinear optimizationminγ,θ{supl[Φ−1(FL(l))− Φ−1(Fm(l, γ, θ))]2}(2.59)which can be numerically evaluated using the Nelder-Mead Algorithm [NM65].2.3.3 LSN ApproximationThe LSN RV constructed by Skew Normal (SN) RV can be used to ap-proximate the sum of the lognormal distribution in Li’s research [LWCW11].A SN distributed RVs SN can be described by PDF fSN (x) and CDFFSN (x) [Azz05]fSN (x) =2ωϕ(x− εω)Φ(αx− εω)(2.60a)FSN (x) = Φ(x− εω)− 2T(x− εω, α)(2.60b)where ϕ(x) and Φ(x) is the standard Gaussian distribution’s PDF andCDF, respectively; the function T (·, ·) is the Owen’s T function; param-eters (α, ω, ε) determine the SN distribution.For the LSN RV LSN = eSN , the corresponding CDF FLSN (x) andPDF fLSN (x) can be derived by Equation (2.60), i.e.,FLSN (y) = FSN (ln(y))= Φ(ln(y)− εω)− 2T(ln(y)− εω, α)(2.61)fLSN (y) =1yfSN (ln(y))=2ωyϕ(ln(y)− εω)Φ(αln(y)− εω).(2.62)For N IID lognormal RVs L1 = eX1 , L2 = eX2 , · · · , LN = eXN , whereXn ∼ N (µ, σ2), a LSN RV LSN can be used to approximate the lognormal202.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVsum Λ, where the critical parameters(α, ω, ε) of LSN can be obtained bymatching the moments between lognormal sum and LSN in the logarithmicdomain. The matching moments include the mean (E[·]), variance (V ar[·]),and skewness (SK[·]).In the logarithmic domain, the three moments of the lognormal sum areobtained using Monte Carlo simulation, i.e.,E[ln(Λ)] = aV ar[ln(Λ)] = bSK[ln(Λ)] = c.(2.63)The three moments of LSN can be calculated using the characteristic of askew normal RV asE[SN ] = ε+ ω√2/πδV ar[SN ] = ω2(1− 2δ2π)SK[SN ] =4−π22π√2π δ3(1− 2 δ2π) 32(2.64)whereδ =α1 + α2. (2.65)Matching moments in the logarithmic domain between Equations (2.63) and(2.64) leads toE[ln(Λ)] ≈ E[ln (LSN)] = E[SN ]V ar[ln(Λ)] ≈ V ar[ln (LSN)] = V ar[SN ]SK[ln(Λ)] ≈ SK[ln (LSN)] = SK[SN ].(2.66)Thus, the parameters (α, ω, ε) of the target LSN RVs areδ =√π2|c|2/3|c|2/3 + ((4− π)/2)2/3 · sign(c)α =δ√1− δ2ω =b1− 2πσ2ε = a− ω√2πδ.(2.67)212.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVHowever, LSN approximation does not guarantee a good approximation,especially when the skewness parameter c in Equation (2.63) is larger than0.995 or LSN is used in tail approximation [LWCW11]. Thus, a transformedSN RV TSN is proposed as TSN = |SN |βsign{SN}, where SN is a skewGaussian RV in Equation (2.60). The corresponding transformed LSN isdefined as TLSN = exp (TSN) = exp (|SN |βsign{SN}). The CDF andPDF of the TLSN areFTLSN (y) = Φ(∣∣∣∣ ln(y)− εω∣∣∣∣1/β · sign{ln (y)− ε})− 2T(∣∣∣∣ ln(y)− εω∣∣∣∣1/β · sign{ln (y)− ε}, α) (2.68)andfTLSN (y) =2| ln (y)|1/β−1ωβy× ϕ(| ln (y)|1/β−1sign{ln (y)} − εω)× Φ(| ln (y)|1/β−1sign{ln (y)} − εω) (2.69)respectively.The three moments of the sum of N lognormal RVs in the transformedlogarithmic domain can be calculated using Monte Carlo simulations asE[| ln(Λ)|1/βsign{ln Λ}]= ςV ar[| ln(Λ)|1/βsign{ln Λ}]= ψSK[| ln(Λ)|1/βsign{ln Λ}]= ϑ.(2.70)After matching moments in Equation (2.70) with the moments in Equation(2.64) in the transformed logarithmic domain, we can obtainE[| ln(Λ)|1/βsign{ln Λ}]≈ E[| ln(Λ)|1/βsign{ln Λ}]= E[SN ]V ar[| ln(Λ)|1/βsign{ln Λ}]≈ V ar[| ln(Λ)|1/βsign{ln Λ}]= V ar[SN ]SK[| ln(Λ)|1/βsign{ln Λ}]≈ SK[| ln(Λ)|1/βsign{ln Λ}]= SK[SN ].(2.71)222.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVThus, the critical parameters can be calculated asδ =√π2|ϑ|2/3|ϑ|2/3 + ((4− π)/2)2/3 · sign(ϑ)α =δ√1− δ2ω =ψ1− 2πσ2ε = ς − ω√2πδ.(2.72)The value of β can be determined by an iterative algorithm proposed by Li[LWCW11].2.3.4 Type-IV Pearson MethodThe Type IV Pearson distribution is shown to have good reproductionof the typical shapes of the sum of the lognormal distribution [NC07], thecorresponding PDF isfZP (z) = ν[1 +(z − µ4µ3)2]−µ1× exp[−µ2 arctan(z − µ4µ3)](2.73)whereµ1 = (r + 2)/2 (2.74)µ2 =−r(r − 2)Skp√16(r − 1)− Sk2p(r − 2)2(2.75)µ3 =√Vp[r − 1− Sk2p(r − 2)2/16] (2.76)µ4 =Mp − (r − 2)Skp√Vp/4 (2.77)ν =[µ3∫ π/2−π/2cosr(x) exp(−µ2x)dx]−1(2.78)r =6(Kup − Sk2p − 1)2Kup − 3Sk2p − 6(2.79)232.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVwhere the parametersMp, Vp, Skp andKup are the mean, variance, skewnessand kurtosis of the Type IV Pearson distribution. When Mp = MT , Vp =VT , Skp = SkT and Kup = KuT , the Type IV Pearson distribution hasthe same statistical characteristics as the sum of the lognormal distribution.They are both convex curves on the lognormal paper [ZS06].However, the Type IV Pearson method has two limitations. First, thevariable domain difference between the sum of lognormal distribution andType IV Pearson distribution. Second, to use Type IV Pearson distributionas the approximation of the sum of lognormal, the selection parameter ofthe sum of lognormal RVs should be less than one. A newly developed TypeIV Pearson variant is introduced to solve these two limitations [CNAG08].Let z(ξ) = ξ− (µ5/ξ)µ6 , µ5 and µ6 are new parameters. Based on the abovemapping, the PDF and CDF of the variant of the Type IV Pearson can beexpressed as follows:gPV (ξ) = z′(ξ)fP [z(ξ)] =[1 +µ6ξ(µ5ξ)µ6]fp[z(ξ)] (2.80)GPV (ξ < A) =∫ A0gPV (ξ)dξ =∫ z(A)−∞fP (z)dz (2.81)wherez(A) = A− (µ5/A)µ6 (2.82)KT =Sk2T (KuT + 3)24(4KuT − 3Sk2T )(2KuT − 3Sk2T − 6). (2.83)When KT < 1, we can use the standard Type IV Pearson distributionto approximate the sum of lognormal distribution and the value of newparameters are shown asµ5 = −11.8− 0.07µ1 + 0.3µ3 + 0.95µ4 (2.84)µ6 = 3.8 + 0.92µ1 − 0.04µ3 + 0.032µ4. (2.85)When KT ≥ 1, we need to approximate the sum of lognormal distributionwith the Type IV Pearson variant.2.3.5 LSG Approximation MethodA log shifted Gamma random, derived from a shifted-Gamma RV, canalso be used to approximate the lognormal sum [LLN07]. The PDF of a242.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVshifted-Gamma RV ZdB isfZdB(z;α, β, δ) =(z − δ)α−1βαΓ(α)exp[−(z − δ)β], z > δ0 , z ≤ δ.(2.86)The corresponding LSG RV is G = 10ZdB/10, whose PDF isfG(g;α, β, δ) =(10 log10 g − δ)α−1ξβαΓ(α)gexp[−10 log10 g − δβ], g > 10δ/100 , g ≤ 10δ/10(2.87)where ξ is ln 10/10. The LSG distribution is determined by three parame-ters: α, β, and δ.For N independent but non-identically distributed Gaussian variables{X1 ∼ N (µ1, σ21), · · · , XN ∼ N (µn, σ2n)}, we use an LSG RV to approximatelognormal sum asΛN =N∑k=1Lk =N∑k=1eXk ≈ Λˆ = eZ (2.88)where ΛN is the lognormal sum; Λˆ is the to-be-determined LSG RV; Z is thecorresponding shifted-Gamma RV. The object is find to the three parameters(α, β, δ) of LSG RV. Similar to the recursive approach in Schwartz and Yeh’sapproximation [SY82], the parameters (α, β, δ) in LSG RV are computedrecursively asΛˆ = eZ ≈ eX1 + eX2 + · · ·+ eXN≈ eZ2 + eX3 + · · ·+ eXN≈ · · ·≈ eZN−1 + eXN .(2.89)Start with the first pair, i.e.,Λ2 = eX1 + eX2 ≈ eZ2 (2.90)where Z2 is a to-be-determined shifted-Gamma RV with parameters (α, β, δ).After matching the mean, log-mean, and log-variance on both sides of the252.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVapproximately equal sign, we have the following equations:E[Λ2] = E[eX1 + eX2 ]Matching======= E[eZ2 ] =eδ(1− β)α (2.91a)E[ln (Λ2)] = E[ln (eX1 + eX2)]Matching======= E[Z2] = δ + αβ (2.91b)V ar[ln (Λ2)] = V ar[ln (eX1 + eX2)]Matching======= V ar[Z2] = αβ2. (2.91c)After defining a new RV Y = X2 − X1 following N (µY , σ2Y ) , we cancalculate the E[ln (Λ2)] and V ar[ln (Λ2)] as followsE[ln (Λ2)] = µX1 +G1(µY , σY ) (2.92a)V ar[ln (Λ2)] = σ2X1 −G21(µY , σY )− 2σ2X1G3(µY , σY )/σ2Y +G2(µY , σY )(2.92b)where G1, G2, and G3 were derived in Schwartz and Yeh’s approximation[SY82]. Based on Equations (2.92) and (2.91), the parameters β and δ canbe expressed in terms of α as:β(α) = α−1√αV ar[ln (Λ2)] (2.93a)δ(α) = −√αV ar[ln (Λ2)] + E[ln (Λ2)]. (2.93b)From the Equations (2.91a) and (2.93), α is the root of following equationg(α):g(α) = δ − α ln (1− β)− ln (E[Λ2]) (2.94)where E[Λ2] = exp (µ1 + σ21/2) + exp (µ2 + σ22/2). Thus, the parameters(α, β, δ) of Z2 can be solved.For the (j − 1)-th iteration in Equation (2.89), i.e., Λj ≈ eZj−1 + eXj ≈eZj , Equations (2.91) changes into the following form:E[Λj ] ≈ E[eZj−1 + eXj ] Matching======= E[eZj ] = eδZj(1− βZj )αZj(2.95a)E[ln (Λj)] ≈ E[ln (eZj−1 + eXj )] Matching======= E[Zj ] = δZj + αZjβZj(2.95b)V ar[ln (Λj)] ≈ V ar[ln (eZj−1 + eXj )] Matching======= V ar[Zj ] = αZjβ2Zj (2.95c)262.3. APPROXIMATING USING A SINGLE NON-LOGNORMAL RVwhere (αZj , βZj , δZj ) are to-be-determined parameters of Zj . We define anew RV W = Zj−1−Xj , which is assumed to be a shifted-Gamma RV. ThePDF of W isfW (w) =∫ ∞−∞fXj (z − w)fZj−1(z)dz=∫ ∞−∞e((z−w)−µj)22σ2jσj√2π(z − δZj−1)αZj−1−1e−(z−δZj−1 )βZj−1Γ(αZj−1)βαZj−1Zj−1dz≈ (w − δW )αW−1e−(w−δW )βWΓ(αW )βαWW.(2.96)From the first moment µW , variance σ2W , and third central moment µ3Wof W [LLN07], the parameters (αW , βW , δW ) can be obtained byαW =4(σ2W )3µ23W=(σ2Xj + αzj−1β2Zj−1)3(αZj−1β3Zj−1)2βW =µ3W2σ2W=αZj−1β3Zj−1σ2Xj + αZj−1β2Zj−1δW = µW − 2(σ2W )2µ3W= δZj−1 + αZj−1βZj−1 − µXj −(σ2Xj + αZj−1β2Zj−1)2αZj−1β3Zj−1.(2.97)Thus, the log-mean E[ln (Λj)] and log-variance V ar[ln (Λj)] areE[ln(eZj−1 + eXj )] = [δZj−1 + αZj−1βZj−1 ] + T1(αW , βW , δW ) (2.98)andV ar[ln (eZj−1 + eXj )]=[αZj−1β2Zj−1 + (δZj−1 + αZj−1βZj−1)2]+ 2T3(αW , βW , δW ) + T2(αW , βW , δW )− µ2Zj(2.99)respectively, where T1(αW , βW , δW ), T2(αW , βW , δW ), and T3(αW , βW , δW )were derived in LSG method [LLN07].Similar to the method of the first pair, eX1 and eX2 in Equation (2.90),βZj and δZj can be expressed in terms of αZj based on Equations (2.95),(2.98), and (2.99). The αZj is the root of equation defined byδZj (αZj )− αZj ln[1− βZj (αZj )]− ln (E[eZj−1 + eXj ]) = 0. (2.100)Thus, the parameters (αZj , βZj , δZj ) are solved.272.4. BOUNDS OF THE SUM OF LOGNORMAL RVS2.4 Bounds of the Sum of Lognormal RVsThe methods listed in the previous sections are based on the assumptionthat a single RV can approximate the sum of lognormal RVs. Anothermethod to analyze the sum of lognormal is bound analysis, which is criticalin analyzing the accurate upper and lower limits of the sum of lognormaland revealing more insights. Meanwhile, bound analysis usually avoids time-consuming calculations.2.4.1 Farley’s Upper BoundFarley’s method can be used to approximate the summands that areindependently and identically distributed. Farley assumed that Y1, · · · , YNare N IID Gaussian RVs with mean µY and standard deviation σY . Theapproximated complementary CDF of the sum of lognormal L =∑Ni=1 Li =∑Ni=1 eYi isP (L > γ) ≈ 1−[1−Q(ln γ − µYσY)]N. (2.101)Beaulieu et al. [BADM94] proved that the Farley’s method is a strict lowerbound for complementary CDF of IID lognormal sum. Accordingly, we canobtain an upper bound as well. This approach is reliable for large variances.Define two exclusive events, A and B, asA = {the event that at least one Li > γ, for i = 1, · · · , t} (2.102a)B = {the complement of event A} . (2.102b)We can obtain the complementary CDFP (L > γ) = P (Lγ,A) + P (L > γ,B)= P (A) + P (L > γ,B)> P (A) = 1− P (Li ≤ γ, i = 1, · · · , N)= 1−[1−Q(ln γ − µYσY)]N.(2.103)The corresponding upper bound of CDF becomesP (L ≤ γ) ≤[1−Q(ln γ − µYσY)]N. (2.104)282.4. BOUNDS OF THE SUM OF LOGNORMAL RVS2.4.2 Slimane’s Upper and Lower BoundsSince Farley’s method can only be used for IID scenarios, Slimane gen-eralized the Farley’s method to independent but not identically distributedscenarios and proposed a more tighter bound on the distribution function[BS01]. After assuming that Y1, , Y2 · · · , YN are N independent but not iden-tically distributed Gaussian RVs with mean µY,n and standard deviationσY,n, Slimane relabeled Y1, , Y2 · · · , YN , such that their labels correspond toarranging them in increasing order, i.e.,Y(1) ≤ · · · ≤ Y(n) ≤ · · · ≤ Y(N). (2.105)Thus, Y(N) = max (Y1, Y2 · · · , YN ) and Y(1) = min (Y1, Y2 · · · , YN ). Usingthe reordered RVs, one can write the sum of lognormal asL = eY(N)[1 + eY(N−1)−Y(N) + · · ·+ eY1−Y(N)] . (2.106)The upper and lower bounds are obtained asLlow = eY(N) ≤ L ≤ Lup = NeY(N) . (2.107)The complementary CDF of L can be written as1− FLlow(x) ≤ P (L ≥ x) ≤ 1− FLup(x) (2.108a)=⇒ 1− FY(N)(lnx) ≤ P (L ≥ x) ≤ 1− FY(N)[ln( xN)]. (2.108b)Since Y1, , Y2 · · · , YN are independent Gaussian RVs, the CDFs of Llow andLup are given asFLlow(x) =N∏n=1[1−Q(ln(x)− µY,nσY,n)](2.109)andFLup(x) =M∏k=1[1−Q(ln(x/N)− µY,nσY,n)](2.110)respectively.Thus, the upper and lower bounds of the complementary CDF of thesum of lognormal RVs areLower : 1−N∏n=1[1−Q(ln(x)− µY,nσY,n)](2.111)292.4. BOUNDS OF THE SUM OF LOGNORMAL RVSandUpper : 1−M∏k=1[1−Q(ln(x/N)− µY,nσY,n)]. (2.112)Note that when Yn are independent and identically distributed GaussianRVs with mean µY and standard deviation σY , the complementary CDF ofLlow becomesP (Llow ≥ x) = 1−[1−Q(ln γ − µYσY)]N(2.113)which is identical to the result in Farley’s method in Equation (2.103).Following Equation (2.106), an improved lower bound can be obtainedusingL ≥ eY(N) + (N − 1)eY(1) . (2.114)Although Y1, , Y2 · · · , YN are independent, Y(1) and Y(N) are correlated.The joint distribution function is given by [BS01]FY(1)Y(N)(x, y) =M∏i=1[1− FYi(Y )], y ≤ xM∏i=1[1− FYi(Y )]−M∏i=1[FYi(y)− FYi(x)], y > x(2.115)whereFYi(x) = Q(y − µYiσYj). (2.116)A tighter lower bound of the complementary CDF isP (L ≥ x) ≥ 1−∫ ln(x)−∞∫ ln[(x−ez)/(N−1)]−∞fY(1)Y(N)(y, z)dydz (2.117)wherefY(1)1Y(N)(y, z) =∂2FY(1)Y(N)(y, z)∂y∂z. (2.118)Using the same procedure, one can obtain a tighter upper bound asL ≤ eY(N) + (N − 1)eY(N−1) . (2.119)302.4. BOUNDS OF THE SUM OF LOGNORMAL RVSThe joint distribution function isFY(N−1)Y(N)(y, z) =N∏i=1[1− FYi(z)] z ≤ y[1−N +N∑i=1FYi(z)FYi(y)]×N∏i=1[1− FXi(y)] z > y.(2.120)A tighter upper bound of the complementary CDF isP (L ≥ x) ≤ 1−∫ ln(x)−∞∫ ln[(x−ez)/(N−1)]−∞fY(N−1)Y(N)(y, z)dydz (2.121)wherefY(N−1)Y(N)(y, z) =∂2FY(N−1)Y(N)(y, z)∂y∂z. (2.122)This bound was adopted to calculate the amount of fading and outage prob-ability [AS02].2.4.3 Tellambura’s Upper and Lower BoundsConsidering Farley’s and Slimane’s methods only deal with independentRVs, Tellambura took previous work one step further. Tellambura’s methodcan be used to calculate the bounds of equally-correlated lognormal sum-mands [Tel08]. Assuming Y1, Y2 · · · , YN are Gaussian RVs as followE[Yn] = µn, n = 1, · · · , N, (2.123a)E[(Yn − µn)2] = σ2Y , n = 1, · · · , N, (2.123b)E[(Yn − µn)(Ym − µm)] = ρσ2Y , m ̸= n, m, n = 1, · · · , N (2.123c)where ρ is the correlation coefficient. Furthermore, {Yn, n = 1, · · · , N} canbe represented asYn = σY√ρZ0 + σY√1− ρZn + µn (2.124)for ρ ≤ 0, n = 1, . . . , N , where Zn(0 ≤ n ≤ N) are N + 1 independent stan-dard Gaussian RVs. Thus, if Z0 = z, {Yn, n = 1, · · · , N} are conditionallyindependent RVs with Yn|z ∼ N [µn + σY√ρ · z, σ2(1 − ρ)]. Averaging overthe distribution of Z0 = z, the CDF of the maximum RVs, we can obtainY(N) in Equation (2.105) asFY(N)(x) =∫ ∞−∞[N∏n=1Φ(x− µn − σY√ρ · zσY√1− ρ)]ϕ(z)dz. (2.125)312.4. BOUNDS OF THE SUM OF LOGNORMAL RVSFor the unimproved upper and lower bound of the sum of lognormal RVs,i.e.,eY(N) ≤ L ≤ NeY(N) , (2.126)the lower and upper bound on the complementary CDF can be obtained bycombining Equations (2.108) and (2.125), resulting inLower : 1−∫ ∞−∞[N∏n=1Φ(x− µn − σY√ρ · zσY√1− ρ)]ϕ(z)dz (2.127)andUpper : 1−∫ ∞−∞[N∏n=1Φ( xN − µn − σY√ρ · zσY√1− ρ)]ϕ(z)dz. (2.128)Note that, if {Yn, n = 1, · · · , N} are independent lognormal RVs (ρ = 0),Equations (2.127) and (2.128) become the Slimane’s result in Equations(2.111) and (2.112), respectively.For the improved lower and upper bound of the sum of lognormal RV,i.e.,eY(N) + (N − 1)eY(1) ≤ L ≤ eY(N) + (N − 1)eY(N−1) (2.129)the lower and upper bounds of the complementary CDF require the condi-tional joint PDF of Y(N) and Y(1) and the conditional joint PDF of Y(N) andY(N−1), respectively. After assuming µi = µ and Z0 = t in Equation (2.124),the conditional joint PDF of Y(N) and Y(1) can be written as [Tel08]fY(1),Y(N)(y, z|Z0 = t) =N !(N − 2)!ϕ(yˆ)ϕ(zˆ)σ2Y (1− ρ)[Φ(zˆ)− Φ(yˆ)]N−2 , y ≤ z(2.130)where Hˆ =H−σY√ρ−µσY√1−ρ . The conditional joint PDF of Y(N) and Y(N−1) canbe written asfY(N−1),Y(N)(y, z|Z0 = t) =N !(N − 2)!ϕ(yˆ)ϕ(zˆ)σ2Y (1− ρ)[Φ(yˆ)]N−2 , y ≤ z. (2.131)Thus, tighter lower and upper bounds of the complementary CDF of a sumof correlated lognormal RVs can be obtained byP (L > x) ≥ 1−∫ ∞−∞∫ lnx−∞∫ ln ym−∞fY(1),Y(N)(y, z|Z0 = t)ϕ(t)dydzdt (2.132)322.4. BOUNDS OF THE SUM OF LOGNORMAL RVSandP (L > x) ≤ 1−∫ ∞−∞∫ lnx−∞∫ ln ym−∞fY(N−1),Y(N)(y, z|Z0 = t)ϕ(t)dydzdt(2.133)whereym =z z < lnxNln(x− ezN − 1)z ≥ ln xN.(2.134)2.4.4 Xiao’s Upper and Lower BoundsXiao et al. proposed global lower and upper bounds for the sum oftwo independent but non-indentically distributed lognormal RVs [XZCW19].Assuming Y1 ∼ N (µ1, σ21) and Y2 ∼ N (µ2, σ22) are two independent GaussianRVs, we can write the CDF of sum of these two lognormal RVs asFL(l) = Pr {L ≤ l}= Pr{eY1 + eY2 ≤ l} (2.135)where the joint PDF of Y1 and Y2 isfY1,Y2(y1, y2) =12πσ1σ2exp(−(y1 − µ1)22σ21− (y2 − µ2)22σ22). (2.136)Thus, the CDF can be rewritten asFL(l) =∫∫Ψ(l)fY1,Y2(y1, y2)dy1dy2 (2.137)where the integral region Ψ(l) is {(y1, y2)| exp (y1)+exp (y2) ≤ l}. The Ψ(l)can also be considered as an infinite point set. The upper and lower boundsof the CDF can be obtained by choosing two new integral regions or pointsets, Ψ−(l) and Ψ+(l), which followΨ−(l) ⊆ Ψ(l) ⊆ Ψ+(l) (2.138)and make Equation (2.137) easy to calculate. Thus, for the CDF FL(l), thelower bound F−L (l) and the upper bound F+L (l) followF−L (l) ≤ FL(l) ≤ F+L (l). (2.139)332.4. BOUNDS OF THE SUM OF LOGNORMAL RVSAfter enlarging and cutting off the initial integral region Ψ(l), Xiao con-structed the lower and upper bounding region of Ψ(l) asΨ−(l) ={(y1, y2)∣∣y1 < y1,m, y2 < y2,m} (2.140)andΨ+(l) ={(y1, y2)∣∣∣∣y2 − y2,m ≤ (−exp (y1,m)exp (y2,m))(y1 − y1,m)}(2.141)where ym = (y1,m, y2,m) is a to-be-determined point that locates on theboundary of Ψ(l), i.e., {(y1,m, y2,m)| exp (y1,m)+exp (y2,m) = l}. Meanwhile,to make the upper and lower bounds of CDF as tight as possible, ym shouldbe the solution to the following two optimization problems:maxymF−L (l,ym)s.t. exp (y1,m) + exp (y2,m) = l(2.142)andminymF+L (l,ym)s.t. exp (y1,m) + exp (y2,m) = l.(2.143)When µ1 →∞ and µ2 →∞, the asymptotic solutions y∗m = (y∗1,m, y∗2,m) ofthe above two optimization problems are identical:y∗m = (y∗1,m, y∗2,m) =(ln(µ1σ22σ22µ1 + σ21µ2l), ln(µ2σ21σ22µ1 + σ21µ2l)). (2.144)Thus, based on Equations (2.137), (2.138), (2.140), and (2.144), thelower bound of the CDF isF−L (l) =∫∫Ψ−(l)fY1,Y2(y1, y2)dy1dy2= Q(µ1 − y∗1,mσ1)Q(µ2 − y∗2,mσ2).(2.145)Based on Equations (2.137), (2.138), (2.141), and (2.144), the upper boundof CDF isF+L (l) =∫∫Ψ+(l)fY1,Y2(y1, y2)dy1dy2= Qµ2 − y∗2,m −exp (y∗1,m)exp (y∗2,m)(µ1 − y∗1,m)√σ22 +(exp (y∗1,m)exp (y∗2,m)σ1)2 . (2.146)342.5. ASYMPTOTIC CDF OF LOGNORMAL SUMAlthough y∗m is an asymptotic solution to Equations (2.142) and (2.143),the obtained lower and upper bounds are applicable in most practical sce-narios [XZCW19].2.5 Asymptotic CDF of Lognormal SumAsymptotic analysis is a kind of approximation that can provide anaccurate estimation of CDF of the sum of lognormal FL(l) as l → 0 orl→∞. Many asymptotic analysis methods come from bound analyses. Thisis because some upper or lower bounds theoretically approach the accurateCDF when l approaches 0 (left tail) or infinity (right tail).2.5.1 Best Lognormal Fit MethodSzyszkowicz and Yanikomeroglu focused on using one RV to approximatethe CDF of lognormal sum in asymptotic scenarios, i.e., the left tail and righttail [SY07]. Since lognormal distributions are mapped onto straight lines inthe lognormal probability paper, the transformation T is defined asT : F (x) 7→ F˜ (x) = Φ−1 (F (ex)) (2.147)where Φ−1 is the inverse CDF of standard Gaussian distribution. If Y ∼N (µ, σ2) is a Gaussian RV, the CDF of the corresponding lognormal RVL = eY isFL(l) = Pr{eY < l} = Φ(ln l − µσ). (2.148)After the transformation T of FL(l), the result is (l − µ)/σ, which is astraight line of strictly positive slope.Defination (Best Lognormal Fit): Given a CDF F (x), after the transfor-mation T in Equation (2.147): F (x) 7→ F˜ (x). Then (x− µ)/σ is consideredto be an asymptote to F˜ (x) at y = ±∞ if and only iflimx→y(F˜ (x)− x− µσ)= 0. (2.149)We can say F (x) has Best Lognormal Fit Φ(lnx−µσ)at ey.Assuming Y1, · · · , YN are N independent but not identically distributedGaussian RVs with mean µY,n and standard deviation σY,n. For the cor-responding lognormal RVs {L1 = eY1 , · · · , LN = eYN }, Szyszkowicz and352.5. ASYMPTOTIC CDF OF LOGNORMAL SUMYanikomeroglu proposed and proved that the asymptotic PDF of L = L1 +· · ·+ LN can be approximated by [SY07]fL(l)l→0≈ ξ√2πσLle− 12(ln l−µLσL)2(2.150)whereσL =(N∑n=11σ2n)− 12(2.151a)µL = lnN + σ2LN∑n=1µnσ2n(2.151b)ξ = NσL(N∏n=1σn)−1(2π)1−N2 e 12σ2L(N∑n=1µnσ2n)2− 12N∑n=1µnσ2n. (2.151c)Szyszkowicz and Yanikomeroglu proved that the PDF in Equation (2.150)has Best Lognormal Fit Φ(ln l−µLσL)at 0. In essence, the authors here usedthe left tail of a lognormal distribution to approximate the left tail pf log-normal sum distribution. However, it was proved that “any lognormal,reciprocal Gamma or log shifted Gamma cannot be used to fit the left tail,under the independence hypothesis” [GXY09].2.5.2 Zhu’s Right Tail Approximation in AsymptoticScenarioSince few studies focus on the right tail of the distribution of lognor-mal sums, Zhu et al. found a closed-form expression for the CDF righttail and proved its tightness. This research focuses on the scenario of twoindependent, but non-identically distributed lognormal RVs.The sum of two lognormal RVs can be expressed as L = eX1+eX2 whereX1 ∼ N (µ1, σ21) and X2 ∼ N (µ2, σ22) are independent and follow Gaussiandistribution. The CDF of L will beFL(l) =∫∫exp(x1)+exp(x2)≤lfX1(x1)fX2(x2)dx1dx2 (2.152)wherefXn(x) =1√2πσ2nexp(−(x− µn)22σ2n)(2.153)362.5. ASYMPTOTIC CDF OF LOGNORMAL SUMfor n = 1, 2.Since the integral region in Equation (2.152) isΘ(x1, x2; l) = {(x1, x2)| exp (x1) + expx2 ≤ l} (2.154)the upper bound of the integral region can be constructed asΘ+(x1, x2; l) = {(x1, x2)| exp (x1) ≤ l, exp (x2) ≤ l}. (2.155)Thus, the upper bound of the CDF in Equation (2.152) isFL(l) ≤ F+L (l) =∫∫exp(x1)≤l,exp(x2)≤lfX1(x1)fX2(x2)dx1dx2= Q(µ1 − ln lσ1)Q(µ2 − ln 2σ2).(2.156)The lower bound of the integral region Θ(x1, x2; l) is constructed asΘ−(x1, x2; l) ={(x1, x2)∣∣e(x1+ϵ) ≤ l, e(x2+ϵ) ≤ l}−{(x1, x2)∣∣x1 ≥ ln (l − (1− e−ϵ)), x2 ≥ ln (l − (1− e−ϵ))}(2.157)where ϵ is a small positive number. The lower bound of the CDF in Equation(2.152) isFL(l) ≥ F−L (l, ϵ) = Q(µ1 − ln l − ϵσ1)Q(µ2 − ln l − ϵσ2)−Q(ln (l(1− exp (−ϵ)))− µ1σ1)×Q(ln (l(1− exp (−ϵ)))− µ2σ2) (2.158)which is related to ϵ.Since ϵ can be arbitrarily small, it is proved that the lower bound F−L (l, ϵ)and the upper bound F+L (l) can be arbitrarily tight as s → ∞ [ZZW+20].The astymptotic CDF of the right tail is1− FL(l) l→+∞≈ Q(ln l − µ1σ1)+Q(ln l − µ2σ2). (2.159)372.5. ASYMPTOTIC CDF OF LOGNORMAL SUM2.5.3 Asymptotic CDF Based on Minimum DistanceTheory and Marcum Q-FunctionCalculating the CDF of the sum of RVs R1, R2, · · · , RN requires inte-grating their joint PDF on the integral region {R1 + R2 + · · · + RN ≤ r},where r is the argument of CDF. Since a lognormal RV Ln can always berepresented by a Gaussian RV Gn, the CDF of the sum of lognormal RVs,{L1, L2, · · · , LN}, can be obtained by integrating the corresponding jointGaussian PDF on the integral region {eG1+eG2+ · · ·+eGN ≤ r}. Thus, theintegral in lognormal probability environment can be converted to a newintegral in Gaussian probability environment, often leading to results in-volving familiar and easy functions, such as the Q-function and the MarcumQ-function. This approach was used in Xiao’s upper and lower bound in theSection 2.4.4 and Zhu’s right-tail approximation in the Section 2.5.2.When calculating CDF in a multivariate Gaussian environment with themean point µG = (µ1, µ2, · · · , µN ), if the original integral region doesn’tcontain µG, we can replace the original integral region with a new integralregion without introducing a significant error in the CDF calculation. Thisnew region should be located inside the initial region and contains the partsof the original region closest to µG, i.e., minimum distance theory [BB99],[BL20]. The integration within the new region is a lower bound of theoriginal. Furthermore, this lower bound rapidly approaches the accurateCDF as the minimum distance between the original integral region and µGincreases [ZCY+18]. When the minimum distance approaches infinite, theerror between the lower bound and the accurate CDF approaches zero. Thus,the lower bound based on the minimum distance theory is an asymptoticsolution to the accurate CDF.In a multivariate Gaussian environment, the integration within a circlecan be represented by the complementary function of Marcum Q-function[SA05]. Thus, minimum distance theory is often combined with MarcumQ-function, resulting in an elegant asymptotic CDF.Beaulieu’s Asymptotic CDF Based on Maximum Disk IntegralRegionBeaulieu and Luan used Marcum Q-function to propose a closed-formasymptotic CDF for the left tail of the sum of two IID lognormal RVs [BL20].Meanwhile, the corresponding Gaussian RVs were assumed to be zero-mean.382.5. ASYMPTOTIC CDF OF LOGNORMAL SUMThe CDF of the sum of two IID lognormal RVs, L1 and L2, is given byFL(l) = Pr{L1 + L2 ≤ l}= Pr{eX + eY ≤ l}= Pr{eX−ln(l) + eY−ln(l) ≤ 1}(2.160)where X and Y are two IID zero-mean Gaussian RVs following N (0, σ2).The Equation (2.160) is the integration in the bivariate Gaussian environ-ment, where the integral region is {(x, y)|ex−ln(l) + ey−ln(l) ≤ 1}. Argumentl will be significantly small when approximating the left tail probability andcalculating the asymptotic CDF. Thus, the integral region can be assumedin the third quadrant of XY Gaussian plane without loss of generality. Thepoint in the integral region with the minimum distance from the mean point,i.e., the origin, is(xm, ym) =(ln(l2), ln(l2))(2.161)which is also called vertex.By the minimum distance theory, the CDF in Equation (2.160) can beapproximated by choosing a new integral region that should contain (xm, ym)and its neighbourhood in the original integral region. In the XY Gaussianplane, the integral over a circular region can be directly represented by aMarcum Q-function. Meanwhile, the original integral boundary is a curvethat can be approximated by a circular boundary. Thus, the object nowis to find an optimal circular integral region to approximate the originalexponential region for each argument l in Equation (2.160). Note that theboundary of the integral region is invariant in shape regardless of how theargument l varies. Changes in the argument l simply translate the bound-ary along the straight line y = x. Thus, to approximate the CDF, if we canfind an optimal circular boundary for a specific original boundary, then thiscircle can be used to approximate all the original boundaries after a propertranslation. The optimal circular boundary was constructed to be tangentto the original boundary at (xm, ym) and has the maximum radius. Sincethe original and circular boundary are shape-invariant for all l, Beaulieu etal. assumed that the center of the circular boundary is the origin. The max-imum radius of a circular region can be maximized by solving the following392.5. ASYMPTOTIC CDF OF LOGNORMAL SUMoptimization problemMaximize rsubject to r =√2 ln(l2)r ≤√x2 + y2l = ex + eyr ≥ 0l ≥ 2.(2.162)The numerical solution is rm = 1.386085392087693. The corresponding lmis 5.329500721943250. This result indicates that if the original boundary is{(x, y)|ex+ey = lm}, the optimal circular boundary is {(x, y)|x2+y2 = r2m}.For the boundary with an arbitrary l, i.e., {(x, y)|ex + ey = l}, its optimalcircular boundary is {(x, y)|(x + ln lm − ln l)2 + (y + ln lm − ln l)2 = r2m}.Thus, the CDF in Equation (2.160) can be written asFL(l) = Pr{L1 + L2 ≤ l}= Pr{eX + eY ≤ l}≥ Pr{(X + ln lm − ln l)2 + (Y + ln lm − ln l)2 ≤ r2m}= Pr{(X + ln lm − ln lσ)2+(Y + ln lm − ln lσ)2≤(rmσ)2}(2.163)where X+ln lm−ln lσ andY+ln lm−ln lσ are two independent, Gaussian distributedRVs with common mean (ln lm − ln l)/σ and unit variance. Thus, theterm(X+ln lm−ln lσ)2+(Y+ln lm−ln lσ)2follows the second-order noncentral chi-squared distribution whose CDF can be expressed as the first-order MarcumQ-function defined as Q1(a, b) =∫∞b x exp(−x2+a22)I0(ax)dx where I0(x)is the modified Bessel function of the first kind with order zero. The a is thenoncentrality parameter defined as a =√((ln lm − ln l)/σ)2 + ((ln lm − ln l)/σ)2,which describes the distance between the center of the circular region inEquation (2.163) and the origin in the bivariate Gaussian environment. Theb indicates the radius of the circular region by b = rm/σ.Thus, the lower bound of the CDF of lognormal sum isFL(l) = Pr{L1 + L2 ≤ l}≥ 1−Q1(√2 (ln (lm)− ln (l))σ,rmσ).(2.164)402.5. ASYMPTOTIC CDF OF LOGNORMAL SUMWhen the distance between the minimum-distance point in Equation (2.161)and the origin approaches infinity, i.e., l → 0, the lower bound in Equation(2.164) rapidly approaches the accurate CDF FL(l):FL(l) ≈ 1−Q1(√2 (ln (lm)− ln (l))σ,rmσ), l→ 0. (2.165)The Equation (2.165) is the asymptotic solution to the CDF of the sum oftwo IID lognormal RVs.Zhu’s Asymptotic CDF of the Sum of Correlated Lognormal RVsBased on Curvature MatchingZhu et al. proposed an asymptotic CDF of the sum of multiple indepen-dent log-normal RVs based on curvature matching [ZCY+17]. Subsequently,they generalized this approach to identically distributed but correlated sce-narios [ZCY+18]. Here, we only introduce the more general approach.For a correlated Gaussian vector g = [G1, · · · , GN ]T , where [·]T denotesthe transpose operation and all the elements have identical mean µ andvariance σ2, g can be expressed by another Gaussian vector x asg = Ax (2.166)where x = [X1, · · · , XN ]T has independent and identically distributed Gaus-sian elements following N (µX , σ2X); the N ×N matrix A isA ≜a 1 1 · · · 11 a 1 · · · 1.......... . ....1 1 1 · · · a (2.167)where a is a parameter which can be calculated from the correlation coeffi-cient ρ between the elements in g. Since the mean, variance, and correlationmatrices on both sides of Equation (2.166) must match each other, the re-lationship between ρ and a followsρ =2a+N − 2a2 +N − 1 . (2.168)Furthermore, µX equalsµa+N−1 and σ2X equalsσ2a2+N−1 .412.5. ASYMPTOTIC CDF OF LOGNORMAL SUMThus, forN correlated and identically distributed lognormal RVs, {L1, · · · , LN} ={eG1 , · · · , eGN }. The CDF of their sum can be written asFL(l) = Pr{L1 + · · ·+ LN ≤ l}= Pr{eG1 + · · ·+ eGN ≤ l}= PrN∑n=1expaXn + N∑k=1,k ̸=nXk ≤ l .(2.169)The integral region isΘ(l) =(x1, · · · , xN )∣∣∣∣∣N∑n=1expaxn + N∑k=1,k ̸=nxk ≤ l . (2.170)To obtain the asymptotic solution to the CDF with the Gaussian in-tegrand, Zhu et al. proved that it is valid to approximate the original in-tegral region with its subset that contains the neighbourhood of the near-est point to the mean of the Gaussian integrand [ZCY+18]. Based on theKarush-Kuhn-Tucker conditions, the nearest point xnst can be calculated as[1a+N−1 ln(lN), · · · , 1a+N−1 ln(lN)]T. Thus, to construct a new integral con-tain xnst and its neighbourhood, Zhu performed a third-order Taylor seriesexpansion of the original boundary at xnst. The zeroth-, first-, and second-order partial derivatives of the new integral region Ψ˜(l) are identical to thefirst-three order partial derivatives of Ψ(l). Note that if we assume two inte-gral regions intersect at xnst (zeroth-order partial derivatives are identical),these two regions must be inner-tangent at this point automatically (first-order partial derivatives are identical). Thus, this method constructs a newintegral region Ψ˜(l) by matching the curvature of the two regions at xnst:Θ˜(l) ={(x1, · · · , xN )∣∣∣∣∣N∑n=1(xn −(ln(lN)a+N − 1 −(n− 1 + a)(1− a)2))2≤((N − 1 + a)√N(1− a)2)2}.(2.171)422.5. ASYMPTOTIC CDF OF LOGNORMAL SUMThus, the asymptotic CDF isFL(l)≈∫Θ˜(l)fx,iid(x)dx= PrN∑n=1(Xn −(ln(lN)a+N − 1 −(n− 1 + a)(1− a)2))2≤((N − 1 + a)√N(1− a)2)2= PrN∑n=1Y 2n ≤((N − 1 + a)√N(1− a)2)2(2.172)where fx,iid(x) is the joint Gaussian PDF of x = [X1, · · · , XN ]T and Yn ∼N(µXσX−(ln (l)−ln (N)(a+N−1)σX −(n−1+a)(1−a)2σX), 1). The∑Nn=1 Y2n followsNth-order non-central chi-squared distribution whose CDF can be expressed as the MarcumQ-function defined as QN (a, b) =∫∞b x(xa)(N−1)exp(−x2+a22)IN−1(ax)dxwhere IN−1(x) is the modified Bessel function of order N − 1. Thus, Equa-tion (2.172) is simplified asFL(l)= 1−QN2(√NσX∣∣∣∣µX − ( 1a+N − 1 ln(lN)− N − 1 + a(1− a)2)∣∣∣∣ ,|N − 1 + a| √N(1− a)2σX).(2.173)When a → ∞, ρ → 0. The correlated lognormal scenario degenerate intoindependent scenario. The result in Equation (2.173) are identical to theresults in [ZCY+17].Zhu’s Asymptotic CDF of the Sum of Two Correlated andNon-Identically Distributed Lognormal RVsZhu et al. also analyzed the asmptotic CDF in corrleated and non-identically distributed scenarios [ZCW+18, ZC19].The correlated and non-identically distributed lognormal random vectoris expressed as [L1, L2]T = [eX1 , eX2 ]T , where [·]T denotes the transposeoperation, and X1 ∼ N(µ1, σ21)and X2 ∼ N(µ2, σ22)are correlated andnon-identically distributed Gaussian RVs. The covariance matrix of X =432.5. ASYMPTOTIC CDF OF LOGNORMAL SUM[X1, X2]T is denoted by R. The joint PDF of X1 and X2 isfX(x) =1√(2π)2det(R)exp(−12(x− µ)TR−1(x− µ))(2.174)where det(·) denotes the determinant and µ = [µ1, µ2]T . Thus, the CDF ofL = L1 + L2 isFL(l) = Pr{eX1 + eX2 ≤ l}=∫∫ex1+ex2≤lfX(x)dx.(2.175)The original integral region Θ(x; l) is {(x1, x2)|ex1 + ex2 ≤ l}. The asymp-totic solution to Equation (2.175) requires the discussion in three cases.Case 1: {[1, 0]R−1µ→ +∞[0, 1]R−1µ→ +∞. (2.176)The solution in Case 1 is similar to the method in Section 2.5.3, i.e., firstobtain the minimum-distance point xm and then construct the approximateintegral region. The xm is defined asxm ≜ argmaxx∈Θ(x;l)fX(x)= argminx∈Θ(x;l)(x− µ)TR−1(x− µ) (2.177)The asymptotic solution to xm isxm,∞ =[x1,m,∞x2,m,∞]=ln( lσ21σ22ρµ2σ1σ2+ρµ1σ1σ2−µ2σ21−µ1σ22 ( ρµ2σ1σ2 − µ1σ21 ))ln(lσ21σ22ρµ2σ1σ2+ρµ1σ1σ2−µ2σ21−µ1σ22(ρµ1σ1σ2− µ2σ22)) (2.178)where ρ is the correlation coefficient between X1 and X2. Based on thecurvature matching, the original integral region Θ(x, l) can be approximatedby a circular integral region Θ˜(x, l) asΘ˜(x, l) ={(x1, x2)∣∣∣ (x1 − x1,c)2 + (x2 − x2,c)2 ≤ κ−2} (2.179)where κ is the curvature of the boundary of the original integral regionΘ(x, l) at xm, which can be calculated asκ =∣∣∣ l exp (x1,m,∞)(l−exp (x2,m,∞))2∣∣∣((l−exp (x2,m,∞))2+exp (2x1,m,∞)(l−exp (x2,m,∞))2) 32(2.180)442.6. COMPARISONS OF METHODSxc = [x1,c, x2,c]T is the center of Θ˜(x, l), i.e.,x1,c = x1,m,∞ −(1 +(exp (x1,m,∞)l−exp (x2,m,∞))−2)− 12κx2,c = x2,m,∞ −(1 +(exp (x1,m,∞)l−exp (x2,m,∞))−2)− 12κ.(2.181)It can be verified that Θ˜(x, l) and Θ(x; l) have identical first- and second-order derivatives at xm.Based on the Marcum Q-function, the asymptotic solution of CDF inEquation (2.175) isFL(l) =(1−Q1(∥∥∥∥xc − xx,∞ + σ1σ2R−1 (xx,∞ − µ)√σ1σ2∥∥∥∥ , κ−1√σ1σ2))× exp(12[∥∥√σ1σ2R−1 (xx,∞ − µ)∥∥2 − (xx,∞ − µ)T R−1 (xx,∞ − µ)])(2.182)where ∥ · ∥ is the 2-norm.Case 2: {[1, 0]R−1µ→ +∞[0, 1]R−1µ→ −∞. (2.183)The asymptotic CDF only depends on the X1 asFL(l) = Q(µ1 − ln (l)σ1)(2.184)Case 3: {[1, 0]R−1µ→ −∞[0, 1]R−1µ→ +∞. (2.185)The asymptotic CDF only depends on the X2 asFL(l) = Q(µ2 − ln (l)σ2). (2.186)2.6 Comparisons of MethodsWe have discussed various approximation methods in the previous sec-tions. Some methods focus on the estimation of the CDF and complemen-tary CDF, while others concentrate on the approximation of the momentsof the sum distribution. In this section, we will compare all the methods.452.6. COMPARISONS OF METHODSApproximation using single lognormal RV: The Wilkinson approach usedthe first two moments matching and gave a good approximation of the sumof lognormal distribution for a limited range of small values of dB spread.By contrast, Schwartz and Yeh’s method also used the first two moments.The approximating result is more accurate and has a broader range of ap-plicability than the Wilkinson method, especially for the practical values ofthe dB spread (6dB ≤ σ ≤ 12dB). Cumulants matching method divided therange into three regions and achieved over ranges of values of the CDF from0.9 to 10−7. However, this method used a generalized Gram-Charlier series,and the series may diverge in some cases. Beaulieu and Xie investigated onthe basis of Wilkinson’s and Schwartz & Yeh’s approximation and developedthe minimax method. This simple method gives a good approximation tothe sum of lognormal RVs in the transformed distribution domain, whereasthe accuracy decreases as the number of RVs increases. MGF approachachieves high approximation accuracy without extremely precise numericalcalculations. Unlike the previous approximation methods, the MGF methodprovides parametric flexibility to accurately approximate different sectionsof the sum of lognormal distribution. In order to achieve high accuracyapproximation, one should choose the matching pairs properly.Approximation using single non-lognormal RV: Inspired by the minimaxmethod, LSQ and curve fitting methods are derived. Two different func-tions were used in these two methods since the linear approximation in theminimax method cannot achieve a desired accuracy when the number of RVincreases. LSQ approach applied linear least squares optimization, whilethe curve fitting method used a nonlinear least squares method for parame-ter optimization. The disadvantage of these two approximation methods isthat numerical computation results are required before further optimization.LSG method also proposed another parametric distribution to approximatebetter when the number of RV increases, and this method provides good ac-curacy over a wide CDF range without the need for LSQ and curve fitting ofnumerical computation. The shortcoming of this method is that the approx-imation is not very accurate in the lower region of CDF for some cases. TheType IV Pearson method motivated by its distribution has the closest PDFand CDF shapes as the sum of the lognormal distribution. The parameterevaluations of the PDF are straightforward, whereas the limitation is thatthe method suits the large N and small σ case. The LSN method can be usedto represent a wide range of lognormal sum RVs and only needs momentsof the target lognormal sum distribution from simulations. The LSN ap-proach has a restriction for the skewness of samples, but the approximationis accurate for a wide range of CDFs.462.6. COMPARISONS OF METHODSTable 2.4: Using a Distribution to Approximate Lognormal Sum Distribu-tion.Author(s) Ref.IndependenceRequiredIdenticallyDistributedSimulationRequiredNumberof RVsWilkinson [Fen60] No No No No LimitSchwartz et al. [SY82] Yes No No No LimitBeaulieu and Xie [BX04] No No All range No LimitNeelesh et al. [MWMZ07] Yes No Part range No LimitSchleher et al. [Sch77] Yes Yes No No LimitZhao and Ding [ZD06] No No All range No LimitBeaulieu and Rajwani [BR04] No No All range No LimitLi et al. [LWCW11] Yes YesRequiremomentsNo LimitChen and Nie [NC07] Yes No No No LimitLam and Le-Ngoc [LLN07] Yes No No No LimitBounds are also used to approximate the sum of the lognormal distribu-tion. Farley’s upper bound gives highly accurate results for CDF less than10−2 with σ = 6 dB and σ = 12 dB, which is better than Schwartz andYeh’s method in this case. Since Farley’s upper bound can only be used forIID scenarios, Slimane generalizes Farley’s bound to independent but notidentically distributed scenarios and this bound is tighter. Since the pre-vious bounds only deal with independent RVs. Tellambura developed thenew upper and lower bounds, which can be used to calculate the bounds ofequally-correlated lognormal summands. The bounds mentioned above arederived using statistical calculation, while Xiao’s upper and lower boundsare proposed by scaling the integration area. This method doesn’t neednumerical calculation, but only works for two RVs.Asymptotic methods focus on the left or right tail of the distribution ofthe lognormal sum, as shown in Table 2.6. Since any lognormal RV cannot beused to fit the left tail of lognormal sum distribution in independent scenarios[GXY09], we will not discuss the Best Lognormal Fit method in Section 2.5.1here. Only Zhu et al. analyzed the asymptotic results for the right tail of thelognormal sum distribution [ZZW+20]. This method works in the scenarioof two independent but not identically distributed lognormal RVs. For theleft tail analysis, all three methods use Marcum Q-Function to approximatethe left tail distribution of the lognormal sum. The method of Beaulieu andLuan only works in the two i.i.d lognormal RVs scenario [BL20]. Zhu et al.derived a asymptotic result for scenario where lognormal RVs are equally472.6. COMPARISONS OF METHODScorrelated and identically distributed [ZCY+18]. The number of lognormalRVs is unlimited. The method of Zhu and Cheng can approximate the CDFof the sum of two arbitrary lognormal RVs asymptotically [ZC19].Table 2.5: Different BoundsAuthor(s) Ref.IndependenceRequiredIdenticallyDistributedTightnessUpperBoundLowerBoundNumercialIntegrationRequiredNumberofRVsFarley [BADM94] Yes Yes Low Yes No No No LimitSlimane [BS01] Yes No Midium Yes Yes No No LimitSlimane [BS01] Yes No High Yes Yes Yes No LimitTellambura [Tel08]EquallycorrelatedNo Medium Yes Yes Yes No LimitTellambura [Tel08]EquallycorrelatedNo High Yes Yes Yes No LimitXiao et al. [XZCW19] Yes No Medium Yes Yes No 2Table 2.6: Different Asymptotic MethodsAuthor(s) Ref.IndependenceRequiredIdenticallyDistributedLeft orRightTailReadilyComputableFunctionNumberofRVsZhu et al. [ZZW+20] yes no Right Q function 2Beaulieuand Luan[BL20] yes yes LeftMarcum Q-Function2Zhu et al. [ZCY+18]Equallycorrelatedyes LeftMarcum Q-FunctionNolimitZhuand Cheng[ZC19] no no LeftMarcum Q-Function248Chapter 3Application of the Sum oflognormal RVs3.1 Co-Channel Interference AnalysisIn cellular radio systems, since the greater radio frequency noise floorand a large number of base stations, interference has become the bottle-neck in increasing capacity. Co-channel interference and adjacent channelinterference are major system interference.Figure 3.1: Frequency reuses of cellular radio systemsCo-channel interference originates from frequency reuse. The frequencyreuse of cellular radio systems is shown in Fig. 3.1, where the whole avail-able frequencies is divided 7 sets and each set is assigned to one cell. Every7 cells using the complete set of frequencies make up a cluster with size of7. In cellular radio systems, some cells between different clusters use thesame set of frequencies. These cells are co-channel cells. For a specific cell,interference from other cells is called co-channel interference. To keep the493.2. DIVERSITY COMBININGco-channel interference level below the tolerance, the most effective methodis separating the co-channel cells by a sufficiently long distance. If the cov-erage of a cell remains unchanged, one way to increase the distance betweenco-channel cells is to increase the cluster size, i.e., the number of cells inone cluster. However, the larger size of a cluster leads to less number ofclusters in a specific region. The number of served users will be limited.Furthermore, unlike thermal noise that can be overcome by increasing thesignal-to-interference ratio, eliminating co-channel interference cannot relyon simply increasing the transmission power. This is because the increase intransmission power will increase the interference to neighboring co-channelcells. Thus, quantitatively describing the magnitude of interference is ofgreat significance to designing and establishing cellular systems.Let i0 be the number of neighboring co-channel cells. The SIR can bewritten asSI=S∑i0i=1 Ii(3.1)where S is the signal power from the desired base station; Ii is the interfer-ence power from the base station in the co-channel cells.Note that co-channel interference Ii can be modeled as a lognormal RV.The∑i0i=1 Ii can be modeled as the sum of lognormal RVs. Thus, the SIRcan be analyzed quantitatively using the methods mentioned in Chapter 2.3.2 Diversity CombiningFading is a phenomenon in which the envelope of a steady and single-frequency radio wave fluctuates over time through the transmitting process.The diversity technique is an effective approach to reducing fading at a rela-tively low cost. The concept of diversity is straightforward. When one radiopath endures severe fading, other independent paths may have stronger sig-nals. We can choose the path with the strong signal, and the instantaneousand average Signal-to-Noise Ratio (SNR)s at the receiver can be improved.The outline of the combining algorithm is shown in Fig. 3.2. Each outputrn from the diversity branch is multiplied by a weight gn. The output of thediversity combining is defined asr =N∑n=1rngn (3.2)where rn contains noise process n(t) and signal sn, i.e., rn = sn+n(t), wherern may follow lognormal distribution. The value of gn varies with the type503.2. DIVERSITY COMBININGof diversity combining algorithms. Three primary combining algorithms areSC , MRC , and EGC . Since MRC and EGC are directly related to the sumof the lognormal RVs, we will only focus on MRC and EGC algorithms inthis section.Figure 3.2: Concept of diversity combining (gn is the weight for eachbranch)3.2.1 MRCThe MRC originates from the criterion of maximized output SNR. Theoutput of the diversity combining can be written asr(t) =N∑n=1gnsn + n(t)N∑n=1gn. (3.3)After assuming that each brand has the same noise power spectral densityN0, we can obtain the instantaneous SNR asSNR =(∑Nn=1 gnsn)2(n(t)∑Nn=1 gn)2 =(∑Nn=1 gnsn)2N0∑Nn=1 g2n. (3.4)Since the criterion is to maiximize the SNR in Equation (3.4), we use513.2. DIVERSITY COMBININGCauchy–Schwarz inequality to rewrite the numerator of the SNR as(N∑n=1gnsn)2≤(N∑n=1g2n)(N∑n=1s2n). (3.5)The Equation (3.5) holds when gn = ksn, where k is a scalar factor. Settingk =√N0, Equation (3.4) becomesSNRmax =∑Nn=1 s2nN0. (3.6)If sn follows a lognormal distribution, i.e., sn = eXn , where Xn ∼ N (µn, σ2n),s2n also follows lognormal distribution. Thus, the output of MRC SNRmaxcan be thought of as the sum of lognormal RVs. All the methods in Chapter2 can be used to analyze the output of MRC. For example, the outageprobability of MRC in lognormal channel can be written asPMRCout (γth) = Pr(N∑n=1exp(2Xn) ≤ γth)(3.7)which is the CDF of the lognormal sum.3.2.2 EGCCompared to MRC, EGC is a more straightforward technique since wedo not need to find the time-varying SNR for each branch. Assuming eachbranch has the same noise power spectral density N0, the output SNR isSNR =1N0N(N∑n=1sn)2. (3.8)Similar to the expression of MRC, if sn follows lognormal distribution, theSNR of EGC can also be thought of as the sum of lognormal RVs. Thus,output SNR isγEGC =1N(N∑n=1exn)2(3.9)where we assume that the power spectral density is 1 without losing thegenerality. The corresponding outage probability of EGC is also the CDFof the sum of lognormal RVs asPEGCout (γth) = Pr(L∑l=1exp(Gl) ≤√Lγth). (3.10)523.2. DIVERSITY COMBININGThe lognormal model is well used in FSO systems for short distances orweak turbulence conditions. Since the diversity technique is also effectivefor improving the channel performance in clear atmospheric optical channels,the sum of lognormal distribution can be used in FSO systems [SC02].53Chapter 4The Optimal MarcumQ-function for theApproximation of the Sum ofTwo Lognormal RVsThe Marcum Q-function can well approximate the complementary CDFof the sum of lognormal RVs, especially near the origin. The core idea ofMarcum Q-function approximation is to use a circular integral region toapproximate the initial exponential integral region which was not easy tocalculate in the Gaussian plane. The approximate accuracy depends onthe selection of the circular region. One method to obtain the optimalcircular region is based on the third-order Taylor series expansion [ZCY+18,ZCY+17]. Another method is solving the optimization problem numericallyto obtain the largest circular region [BL20]. In this chapter, first, in thescenario of two lognormal RVs with identical and independent distribution,we use geometric methods to analytically solve the optimization problem ofthe maximum radius of the circular region. The obtained maximum radiusis greater than the previous numerical optimization result and is consistentwith the result from the third-order Taylor series. Then, in the scenarioof two correlated lognormal RVs with identical distribution, we prove thatthe maximum radius of the circular region agrees with the prior result in[ZCY+18].4.1 Sum of Two Independent Lognormal RVsIn this section, we study the case of the sum of two lognormal RVswith identical and independent distribution. Let G1 and G2 be independentGaussian RVs with common mean µ and common variance σ2 and define544.1. SUM OF TWO INDEPENDENT LOGNORMAL RVStwo zero-mean Gaussian RVs asX1 = G1 − µ (4.1a)X2 = G2 − µ (4.1b)and the corresponding independent lognormal RVs asS1 = eG1 = eX1+µ (4.2a)S2 = eG2 = eX2+µ. (4.2b)Thus, the CDF of the sum of two independent lognormal RVs, i.e., S =S1 + S2, isFS(s) = Pr {S ≤ s} = Pr {S1 + S2 ≤ s}= Pr{eX1+µ + eX2+µ ≤ s}= Pr{eX1+µ−ln s + eX2+µ−ln s ≤ 1}= Pr{eX1−lnseµ + eX2−lnseµ ≤ 1}=∫∫ΘexpfX1,X2(x1, x2)dx1dx2(4.3)wherefX1,X2(x1, x2) =12πσ2exp(− 12σ2(x21 + x22))(4.4)is the joint PDF of two zero-mean independent Gaussian RVs, X1 and X2.Meanwhile,Θexp(x1, x2; s) ={(x1, x2)|ex1−ln seµ + ex2−ln seµ ≤ 1}(4.5)is the integral region, which we will refer to as the exponential region. Fur-thermore, in the X1 −X2 Gaussian plane, the boundary of the exponentialregion isΓexp(x1, x2; s) ={(x1, x2)|ex1−ln seµ + ex2−ln seµ = 1}(4.6)which we will refer to as the exponential boundary.The Equation (4.6) shows that the boundary in the X1 −X2 Gaussianplane is invariant in its shape, that is, the values of s in CDF and the mean µdon’t change the shape of the boundary, only translate the boundary along554.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSthe straight line x1 = x2. (First Characteristic of Boundary: Shapeinvariance of the integral boundary for independent lognormal RVs)Furthermore, larger µ or smaller s will translate the boundary to negativeinfinity along the line x1 = x2. Lettings′ =seµ, (4.7)the exponential boundary of different s′ in the X1 − X2 Gaussian plane isshown in Fig. 4.1. The point of the mean of (X1, X2) is located at the origin,i.e., (0, 0). Since our objective here is to analyze the left tail probability ofthe sum of lognormal RVs, s′ is usually smaller than 1. The nearest pointon the exponential boundary to the origin isx1 = x2 = lns′2. (4.8)We refer to this closest point as the boundary vertex.Although the exponential region Θexp in Equation (4.3) is shape invariantand is symmetric about the line x1 = x2, integrating fX1,X2(x1, x2) in Θexpis still time-consuming and resource-consuming. Note that in a multivariateGaussian environment such as the X1 − X2 plane in Fig. 4.1, the entiretarget integral region can be approximated by a specific region that locatesinside the target region and contains the parts of the target region that isclosest to the origin [BB99]. Specifically, if an integral region in Fig. 4.1 iscovered by the exponential region Θexp, contains the boundary vertex and itsneighborhood, and makes Equation (4.3) easy to calculate, this new regioncan be a good approximation of Θexp. The integration of fX1,X2(x1, x2) inthe new region is a lower bound to Equation (4.3).Note that the integration of a 2-dimensional Gaussian PDF within acircular region can be easily described by a first-order Marcum Q-function[BL20], which is defined asQ1(α, β) =∫ ∞βx exp(−x2 + α22)I0(αx)dx (4.9)where I0(·) is the modified Bessel function of the first kind with order zero.The parameters α and β describe the position and the radius of the circularregion, respectively. Thus, we turn now to find the optimal circular regionto approximate the initial exponential region. This optimal circular regionneeds to meet the following requirements: (1) The exponential region en-tirely covers the optimal circular region; (2) The optimal circular boundary564.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSFigure 4.1: The boundary of integration region in the X1 −X2 plane, fors′ = 0.25, s′ = 1, s′ = 5.intersects with the exponential boundary at the boundary vertex; (3) Theoptimal circular region has the maximum radius, i.e., rmax, compared withother circular regions.Due to the shape invariance of exponential boundary, the radius of theoptimal circular boundary is invariant. We can fix the center of the circularregion at the origin without losing generality, as shown in Fig. 4.2. Thecircular boundary and the exponential boundary should intersect at theboundary vertex with the location determined as(xv, yv) =(lns′2, lns′2)=(r√2,r√2). (4.10)Meanwhile, the relationship between s′ and r can be described asr =√2 lns′2⇐⇒ s′ = 2e r√2 . (4.11)574.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSFigure 4.2: Using a circular region to approximate exponential regionOur objective is to find the maximum radius rmax and the correspondings′m. Since the boundary vertex in Fig. 4.2 is located in the first quadrant, werewrite the first quadrant’s exponential and circular boundaries as explicitfunctions:yexp(x; r) = ln (s′ − ex) = ln (2e r√2 − ex) (4.12)andycir(x; r) =√r2 − x2 (4.13)respectively. The difference between the exponential and circular boundaryfunctions isy∆(x; r) = yexp(x; r)− ycir(x; r) = ln (2er√2 − ex)−√r2 − x2. (4.14)Then, the first derivative of exponential and circular boundaries arey˙exp(x; r) = − exs′ − ex = −ex2er√2 − ex(4.15)584.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSandy˙cir(x; r) = − x√r2 − x2 (4.16)respectively. The difference between the first derivatives of exponential andcircular boundaries isy˙∆(x; r) = y˙exp(x; r)− y˙cir(x; r) = − ex2er√2 − ex+x√r2 − x2 . (4.17)At the boundary vertex in Equation (4.8),y˙exp(xv; r) = y˙cir(xv; r) = −1. (4.18)The Equation (4.18) shows that as long as the exponential and circularboundaries intersect at the boundary vertex (xv, yv), these two boundariesmust be tangent at this point, no matter what s′ or r is. (Second Charac-teristic of Boundary: Tangency invariance at the vertex for independentlognormal RVs)After obtainingy∆(xv; r) = yexp(xv; r)− ycir(xv; r) = 0 (4.19)andy˙∆(xv; r) = y˙exp(xv; r)− y˙cir(xv; r) = 0 (4.20)we analyze the second derivative of boundary functions and define the secondderivative of the difference between two boundary functions asy¨∆(x; r) =d2dx2(yexp(x)− ycir(x))= − 2exer√2(2er√2 − ex)2 + r2(r2 − x2) 32.(4.21)At the boundary vertex (xv, yv) = (r√2, r√2), the value of Equation (4.21)isy¨∆(xv; r) = − 2er√2 er√2(2er√2 − e r√2)2 + r2(r2 − ( r√2)2) 32= −2 + 2√2r.(4.22)594.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSThus, we can obtain the positive and negative of y¨∆(xv; r) with differentvalues of the radius r, i.e.,y¨∆(xv; r)> 0, r <√2;= 0, r =√2;< 0, r >√2.(4.23)Based on Equations (4.19) and (4.20), if y¨∆(xv; r) < 0, y˙∆(xv + ϵ1; r) < 0and y∆(xv+ ϵ2; r) < 0, where we assume that ϵ1 and ϵ2 are sufficiently smalland 0 < ϵ1 < ϵ2 without losing generality. The y∆(xv + ϵ2; r) < 0 meansthat the exponential region cannot cover the circular region entirely. Thus,the maximum radius rmax of the circular region can be never greater than√2.The Figure 4.3 shows the intersection of two boundaries when r >√2.Besides the boundary vertex, two additional intersection points exist. Twoparts inside the circular region are beyond the exponential boundaries.When r <√2, we can prove that the exponential and circular boundariesonly intersect at the boundary vertex by plotting the function y∆(x; r), asshown in Fig. 4.4. Since y∆(x; r) always take non-negative values, theexponential region entirely covers the circular region.Finally, the radius rmax of the circular region is√2. The correspondingvalue of s′ in Equation (4.11) is 2e. That is, when the exponential boundaryin Equations (4.6) and (4.7) isΓexp,0 ={(x1, x2)∣∣ex1−ln 2e + ex2−ln 2e = 1} (4.24)the optimal approximate circular boundary isΓcir,0 = {(x1, x2)|x21 + x22 = 2} (4.25)as shown in Fig. 4.2. The Γexp,0 can serve as a reference boundary. Based onthe First Characteristic of Boundary, a new exponential boundary can be ob-tained by shifting the reference boundary Γe,0 by a distance√2(ln 2e−ln seµ )along the negative direction of x1 = x2. Thus, for an arbitrary exponentialregion with different s and µ in Equation (4.5), the corresponding optimalapproximate circular region isΘcir ={(x1, x2)∣∣∣ (x1 + ln 2e− ln seµ)2+(x2 + ln 2e− ln seµ)2 ≤ 2} .(4.26)604.1. SUM OF TWO INDEPENDENT LOGNORMAL RVSFigure 4.3: Exponential region cannot cover the circular region if r >√2.Using the optimal circular region, we rewrite Equation (4.3) asFS(s) = Pr{eX1−lnseµ + eX2−lnseµ ≤ 1}≈ Pr{(X1 + ln 2e− ln seµ)2+(X2 + ln 2e− ln seµ)2 ≤ 2}= Pr{2∑i=1(Xi + ln 2e− ln seµ)2 ≤ 2}= Pr{2∑i=1Y 2i ≤2σ2}(4.27)where Yi ∼ N ( ln 2e+µ−ln sσ , 1), and∑2n=1 Y2i follows the second-order non-central chi-squared distribution whose CDF can be expressed in terms ofEquation (4.9). Thus, we can obtain the approximate CDF of the sum of614.2. SUM OF TWO CORRELATED LOGNORMAL RVSFigure 4.4: The difference between exponential and circular region whenr <√2two independent lognormal RVs using Marcum Q-function asFS(s) ≈ 1−Q1(√2σ(ln 2e+ µ− ln s) ,√2σ). (4.28)Furthermore, this approximation rapidly becomes more accurate as the cir-cular region is further away from the origin, i.e., µ is larger or s is smaller[ZCY+18].The Figure 4.5 shows the probability distribution left tail of the sum oftwo independent lognormal RVs using different methods when σ = 2. Theproposed Marcum Q-method is always more accurate than the previousone [BL20] . This is because the radius of circular region in the previousMarcum Q-method is 1.386085392087693, which is less than the radius√2 inthe proposed method. A larger circular integral region can more accuratelyapproximate the original integral region in Equation (4.3).4.2 Sum of Two Correlated Lognormal RVsIn this section, we study the case of the sum of two correlated lognormalRVs with identical distribution. Let N1 and N2 are two correlated GaussianRVs with common mean µ, common variance σ2, and correlation coefficientρ. Using the method that two correlated Gaussian RVs can be represented624.2. SUM OF TWO CORRELATED LOGNORMAL RVSFigure 4.5: The probability distribution left tail of the sum of two inde-pendent lognormal RVs using simulation and two Marcum Q-methodsby two identical and independent Gaussian RVs [ZCY+18], we rewrite N =(N1, N2)T asN =(N1N2)=(a 11 a)·(X1 + µXX2 + µX)(4.29)where a is a parameter that can be calculated from ρ. Assuming X1 andX2 are two independent and identically distributed Gaussian RVs with zeromean and variance σX . After matching the means, variance, and correlatedmatrices from both sides of Equation (4.29), we can obtainµX =µ1 + a(4.30a)σ2X =σ21 + a2(4.30b)ρ =2a1 + a2. (4.30c)The relationship between a and ρ in Equation (4.30c) is shown in Fig.4.6. When a ∈ (−1, 1), we can establish an one-to-one mapping between634.2. SUM OF TWO CORRELATED LOGNORMAL RVSFigure 4.6: The curve of the relationship between a and ρ in Equation(4.30c)a ∈ (−1, 1) and ρ ∈ (−1, 1). It should be noted that the one-to-one mappingrelationship constructed by the method in Equations (4.29) and (4.30c) islimited when the number of RVs is greater than 2. This is because in thescenario of three or more random variables, ρ cannot take some negativevalue near −1 regardless of the value of a.Using Equation (4.29), we can obtain the CDF of the sum of two corre-lated lognormal RVs:FS(s) = Pr{eN1 + eN2 ≤ s}= Pr{ea(X1+µX)+(X2+µX) + e(X1+µX)+a(X2+µX) ≤ s}= Pr{ea(X1+µX)+(X2+µX)−ln s + e(X1+µX)+a(X2+µX)−ln s ≤ 1}= Pr{ea(X1+µX−11+aln s)+(X2+µX− 11+a ln s)+e(X1+µX−11+aln s)+a(X2+µX− 11+a ln s) ≤ 1}.(4.31)For the a ∈ (−1, 1) calculated from a given ρ ∈ (−1, 1), the values of µX ands will not affect the shape of the boundary. It only translates the boundary.(Third Characteristic of Boundary: Shape invariance of the integralboundary for correlated lognormal RVs)The exponential boundary can be written asΓexp,cor ={(x1, x2)∣∣∣eax1+x2 + ex1+ax2 = s′cor} (4.32)644.2. SUM OF TWO CORRELATED LOGNORMAL RVSwhere s′cor = s · e−(a+1)µX . The exponential boundary with different s′corand a is shown in Fig. 4.7. The intersection between Γexp,cor and x1 = x2,i.e., the boundary vertex is(x1,o, x2,o) =(1a+ 1lns′cor2,1a+ 1lns′cor2)(4.33)Figure 4.7: Γexp,cor with different a and s′corHaving obtained the obtained exponential boundary and the correspond-ing boundary vertex, we need to find the optimal circular domain to approx-imate the integral domain in Equation (4.31). Similar to the method usedin the independent lognormal scenario, we first fix the center of the circulardomain at the origin and then try to find the optimal circular boundarywith maximum radius and corresponding value of s′cor. The optimal circularboundary and the exponential boundary should intersect at the boundaryvertex in Equation (4.33). Thus, the relationship between s′cor and radius r654.2. SUM OF TWO CORRELATED LOGNORMAL RVSisr =√2a+ 1lns′cor2⇐⇒ s′cor = 2e(a+1)r√2 . (4.34)Based on the derivation rule of implicit function, the first derivative ofexponential and circular boundaries arex˙2,exp = −aeax1+x2 + ex1+ax2eax1+x2 + aex1+ax2(4.35)andx˙2,cir = − x1√r2 − x21(4.36)respectively. At the boundary point in Equation (4.33), x˙2,exp = x˙2,cir = −1.Thus, even if the two lognormal RVs are correlated, as long as exponentialand circular boundaries intersect at the boundary vertex, these two bound-aries must be tangent at this point. (Fourth Characteristic of Bound-ary: Tangency invariance at the vertex for correlated lognormal RVs)We define the second derivative of the difference between two boundariesasx¨2,∆(x1; r) =ddx1(x˙2,exp − x˙2,cir) . (4.37)At the boundary point in Equation (4.33), we havex¨2,∆(x1,o; r) = −2(1− a)21 + a+2√2r. (4.38)Since the boundary difference and its first derivative are 0 at the boundaryvertex, x¨2,∆(x1,o; r) must be greater than 0 to keep exponential region cov-ering the circular region. Thus, we can obtain the maximum radius of thecircular regionrmax =√2(1 + a)(1− a)2 . (4.39)The corresponding value of s′cor is 2 · exp[(1+a1−a)2].Thus, whenΓexp,cor,0 ={(x1, x2)∣∣∣eax1+x2 + ex1+ax2 = 2 · e( 1+a1−a )2}={(x1, x2)∣∣∣ea(x1−D)+(x2−D) + e(x1−D)+a(x2−D) = 1} (4.40)where D = 11+a ln 2e( 1+a1−a )2, the optimal approximate circular boundary isΓcir,cor,0 ={(x1, x2)∣∣∣x21 + x22 = 2(1 + a)2(1− a)4}. (4.41)664.2. SUM OF TWO CORRELATED LOGNORMAL RVSFigure 4.8: Exponential boundary Γexp,cor,0 and the corresponding optimalcircular boundary Γcir,cor,0 when a takes different valuesThe Figure 4.8 shows Equations (4.40) and (4.41) at different values ofa. Thus, Γexp,cor,0 can serve as a reference boundary. Based on the ThirdCharacteristic of Boundary, a new exponential boundary can be obtainedby shifting the reference boundary Γexp,cor,0 by a distance dcor asdcor =√2(1 + a(1− a)2 −11 + alns · e−(1+a)µX2)(4.42)along the negative direction of x1 = x2. Thus, for an arbitrary exponentialregion with different s and µX in Equation (4.32), the corresponding optimalapproximate circular region isΓcir,0 ={(x1, x2)∣∣∣ (x1 + dcor√2)2+(x2 +dcor√2)2=2(1 + a)2(1− a)4}. (4.43)674.3. SUMMARYUsing this approximate circular region, we can rewrite the CDF of thesum of two correlated lognormal RVs asFS(s) = Pr{eN1 + eN2 ≤ s}= Pr{(X1 +dcor√2)2+(X2 +dcor√2)2≤ 2(1 + a)2(1− a)4}= Pr{2∑i=1(Xi +dcor√2)2≤ 2(1 + a)2(1− a)4}= Pr{2∑i=1Y 2i ≤2(1 + a)2σ2X(1− a)4}(4.44)where Yi ∼ N ( dcor√2σX , 1), and∑2n=1 Y2i follows the second-order noncentralchi-squared distribution whose CDF can be expressed in terms of Equa-tion (4.9). Thus, we can obtain the approximate CDF of the sum of twocorrelated lognormal RVs using the Marcum Q-function asFS(s) ≈ 1−Q1(√2σX(1 + a(1− a)2 + µX −11 + alns2),√2(1 + a)σX(1− a)2)(4.45)which agrees the prior result in [ZCY+18]. Furthermore, the approximationaccuracy increases rapidly as the circular region is further away from theorigin, i.e., µX is larger or s is smaller [ZCY+18].4.3 SummaryIn this chapter, we introduced a geometric method to analytically cal-culate the optimal circular region for approximating the CDF of the sumof two IID lognormal RVs via the Marcum Q-function. Compared with thenumerical optimization result in the previous work, our analytical resultsimprove the accuracy of the Marcum Q approximation. In the scenario oftwo correlated lognormal RVs proved that the maximum radius of the cir-cular region agrees with the prior results obtained by the third-order Taylorseries expansion.68Chapter 5Conclusion5.1 SummaryIn this thesis, we focused on academic research about the sum of lognor-mal RVs, which is widely used in performance analysis in communicationchannels but doesn’t have a closed-form PDF and CDF.First, we provided a comprehensive survey for the methods of approxi-mating the sum of lognormal RVs. Although lots of methods were proposedto approximate lognormal sums, all methods can be classified into four cate-gories: (1) Using a single lognormal RV to approximate the lognormal sum;(2) Using a different complex RV to approximate the lognormal sum; (3)Bound analysis; (4) Asymptotic analysis. The research on the approxima-tion of lognormal sums starts from the assumption that a lognormal sum canbe considered a single lognormal RV. This assumption introduces a classicmethod to approximate lognormal sums. However, the methods in the firsttwo categories often fail near the origin. Thus, some closed-form asymptoticanalysis was proposed recently.Second, we introduced the scenarios where the lognormal sum is usuallyused. Since lognormal distribution often describes the large-scale fading inthe RF communication channel or the weak turbulence in the free spaceoptical communication channel, lognormal sums are often used to describethe distribution of SNR to analyze the channel performance.Third, in the scenario of two IID lognormal RVs, we proposed a geometricmethod to analytically calculate the optimal circular integral region whenapproximating the CDF of the lognormal sum via a Marcum Q-function.This result improved the approximation accuracy based on the maximumcircular integral region. In the scenario of two correlated lognormal RVs, weproved that the maximum radius of the circular region agrees with the priorresults obtained by the third-order Taylor series expansion.695.2. FUTURE WORK5.2 Future WorkAlthough considerable research on the approximation of lognormal sumhas been proposed during the past decades, more accurate and easy-to-useapproximation methods or bounds are still needed. According to our survey,we now provide two potential directions for the lognormal-sum approxima-tion method. First, when calculating the CDF of the sum ofN IID lognormalRVs (The value of N is arbitrary), the CDF can be theoretically simplified toa single integration due to the symmetry of the integral region in a multiple-Gaussian environment. But this method requires solving a transcendentalequation. Second, based on the semi-infinite rectangle bounds [XZCW19],more accurate upper and lower bounds can be analyzed by the division ofintegral region [ZZCB18].70Bibliography[AS02] M.-S. 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Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Cheng, Julian
A survey of the sum of lognormal distribution and some recent results
Text