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On strong approximations for some classes of random iterates Merlevède, Florence
Description
his talk is devoted to strong approximations in the dependent setting. The famous results of Koml\'os, Major and Tusn\'ady (1975-1976) state that it is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random variables in ${\mathbb L}^p$ ($p >2$) by a Wiener process with an error term of order $o(n^{1/p})$. In the case of functions of random iterates generated by an iid sequence, we we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb L}^\infty$ or in ${\mathbb L}^1$), under which the strong approximation result holds with rate $o(n^{1/p})$. The proof is an adaptation of the recent construction given in Berkes, Liu and Wu (2014). As we shall see our conditions are well adapted to a large variety of models, including left random walks on $GL_d({\mathbb R})$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains. We shall also provide some examples showing that our ${\mathbb L}^1$-coupling condition is in some sense optimal. This talk is based on a joint work with J. Dedecker and C. Cuny.
Item Metadata
| Title |
On strong approximations for some classes of random iterates
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| Creator | |
| Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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| Date Issued |
2017-05-30T09:48
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| Description |
his talk is devoted to strong approximations in the dependent setting. The famous results of Koml\'os, Major and Tusn\'ady (1975-1976) state that it
is possible to approximate almost surely the partial sums of size $n$ of i.i.d. centered random
variables in ${\mathbb L}^p$ ($p >2$) by a Wiener process with an
error term of order $o(n^{1/p})$. In the case of functions of random iterates generated by an iid sequence, we
we shall give new dependent conditions, expressed in terms of a natural coupling (in ${\mathbb L}^\infty$ or in ${\mathbb L}^1$), under which the strong approximation result holds with rate
$o(n^{1/p})$. The proof is an adaptation of the recent construction given in Berkes, Liu and Wu (2014).
As we shall see our conditions are well adapted to a large variety of models, including left random
walks on $GL_d({\mathbb R})$, contracting iterated random functions, autoregressive Lipschitz processes, and some ergodic Markov chains.
We shall also provide some examples showing that our ${\mathbb L}^1$-coupling condition is in some sense optimal. This talk is based on a joint work with J. Dedecker and C. Cuny.
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| Extent |
41.0
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| Subject | |
| Type | |
| File Format |
video/mp4
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| Language |
eng
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| Notes |
Author affiliation: University Paris Est Marne-la-Vallée
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| Series | |
| Date Available |
2019-03-11
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| Provider |
Vancouver : University of British Columbia Library
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| Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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| DOI |
10.14288/1.0376735
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| URI | |
| Affiliation | |
| Peer Review Status |
Unreviewed
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| Scholarly Level |
Faculty
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| Rights URI | |
| Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International