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Stochastic models of changes in population distribution among categories Gerchak, Yigal

Abstract

There are very many processes in the natural and social sciences which can be represented as a set of flows of objects or people between categories of some kind. The Markov chain model has been used in the study of many of them. The basic form of the Markov chain model is, however, rarely adequate to describe social, occupational and geographical mobility processes. We shall therefore discuss a number of generalizations designed to introduce greater realism. In Chapter I we formulate and investigate a general model which results from relaxing the assumptions of sojourn-time's memorylessness and independence of origin and destination states, and of population homogeneity. The model (a mixture of semi-Markov processes) is then used in two ways. First, it provides a framework in which various special cases (which correspond to models which were used by social scientists) can be analytically compared. We pay particular attention to comparisons of rate of mobility in related versions of various models and to compatability of popular parametric forms with observed mobility patterns. Second, any result obtained for the general model can be specialized for the various cases and subcases. In Chapter II we formulate a system-model allowing interaction among individuals (components), which has been motivated by Conlisk. We define processes on this model and analyze their properties. A major effort is then devoted to establishing that when the population size becomes large, this rather complex stochastic model can be approximated by a single deterministic recursion due to Conlisk (1976). Nevertheless, we draw attention to certain aspects (particularly steady-state behavior) in which the approximation may fail. In Chapter III we address ourselves to the issue of measurement of (what we refer to as) social inheritance in intergenerational mobility processes. We distinguish between various aspects and concepts of social inheritance and outline the implications that certain "social values" may have on constructing a measure (or index). In the mathematical discussion which follows certain mechanisms for generating "families" of measures are indicated, and the properties of some particular combinations are investigated.

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