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A Unified Approach to Limit Theorems in Probability Theory and Its Application

K. Nidhin

Abstract



In this paper, we show a convergence behavior of a general class of Borel functions of random variables. Our motivation for this paper is to use this convergence behavior to unify some of the existing important limit theorems, which are available in the literature as independent theorems. To achieve this we derive a general limit theorem in a function space of probability distributions. The special cases of the main theorem includes generalized
central limit theorem, extremal types theorem and limit theorems of random sum, random maxima and random minima. We derive the limit theory of extremes under nonlinear normalization as a special case of the main result. We show the limit theory of product of i.i.d.r.v of fixed sample sizes and random sample sizes as special cases of the main theorem. Moreover, we show how the main theorem can be useful to derive some of the new
limit theorems in probability.

Keywords


central limit theorem, extremal types theorem, stability, weak convergence, convergence of random number of random variables.

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