对数似然比与离散随机变量序列强极限定理的一种分析方法

THE LOGARITHMIC LIKELIHOOD RATIO AND ANANALYTIC APPROACH OF STRONG LIMIT THEOREMS FORTHE SEQUENCES OF DISCRETE RANDOM VARIABLES

本文引进对数似然比作为任意离散随机变量序列相依性的一种度量，并通过限制似然比给出样本空间的某种子集，在这种子集上得到了离散随机变量序列的一类强极限定理，它包含若干经典强大数定律为其特例．在证明中本文提出了证明强极限定理的一种分析方法，其要点是将关于单调函数可微性的定理应用于几乎处处收敛的研究．

In this paper the notion of logarithmic likelihood ratio,as measure of dependence of a sequence of arbitrary discrete random variables,is intreduced.A subset of thesample space is given by restricting the logarithmic likelihood ratio,and on this subset aclass of strong limit theorem for the sequences of discrete random variables,including someclassical strong laws of large numbers as corollaries,are obtained.In the proof an analytictechnique to prove the strong limit theorems is presented, the crucial part of which is theapplication of Lebesgue's theorem on differentiability of monotone functions to the study ofa.e.convergence.