Strong Law Of Large Numbers For Arbitrary Random Sequence

One of the important problems in the field of probability limit theory is the strong limit theorems. The limit theory covers strong convergence of partial sums of random variable and the limist of arithmetic average of random variables.Developments of modern probability limit theory is still in a rapidly evolving,especially when it combines with the other branches of probability and statistics. It playes a very important role in the development of probability and statistics. The limit theory of dependent random sequence is widely used in the field of statistics,insurance, finance, complexity and reliability.This article dedicated to the research of the strong laws for dependent random variables. By using the properties of slowly varying functions, together with the classical Borel-Cantelli lemma in probability and pure analytical method, and the concept of randomized controlled random variables, the strong convergence for dependent random variables has been studied, and a sufficient condition for general founded strong law of large numbers arbitrarily dependent random sequence is obtained and several sufficient conditions for the case of any dependent weighted strong laws are alse obtained, The results we obtained are generalize the existing results. This paper is divided into six chapters:The first chapter introduces the main part of the research background, main content and intuitive background and the classical law of large numbers, and some research method used in this paper.The second chapter includ the basic theory and concepts of several law of large numbers and related knowledge. Some existing results of the strong limit theorems for random sequences are given.In the third chapter, on the basis of previous studies by using the properties of slowly varying function, the Borel-Cantelli lemma and pure analysis method, some sufficient conditions for strong law of large numbers are established with dependent but different distributions of random variable.In the fourth chapter, we continue to put the concept of random contraled into the research of the weighted sums of the random variables. By means of the properties of truncation and the common method of partition in probability, strong convergence of weighted sums of arbitrary random sequence are obtained.In the fifth chapter, on the basis of chapter III chapter IV, and using the classical method of moments, the strong convergence of the moment inequality and random sequence are established.In the sixth chapter, we summarize the full text, and some future research work are forward.