On Borel-Cantelli Lemmas For Random Variables Sequence And Their Applications

Probability theory is a mathematics branch which is applied to study the random phenomena and quantity laws. Since twentieth Century, promoted by physics, biology, engineering and technology (such as automation, radio technology), probability theory has achieved rapid development. Especially as new methods and new tools appear continuously in recent years, the scope of application of the probability theory has been greatly expanded, and the methods of probability theory have also been introduced to various social sciences and engineering disciplines, such as, demographics, economic decision, earthquake prediction and weather forecasting, product inspection and quality control and so on. At the same time, the need of the actual problem effectively promotes the rapid development of probability theory in turn. Probability limit theory, as one of main branches of probability theory, also obtained the full development. The research contents of classical limit theory have been greatly expanded from the earliest on the partial sum of a sequence of independent random variables to various dependent sequences and martingale sequences.This study is mainly on Borel-Cantelli lemma which is a very important lemma in the probability theory. It is the main tool which proves that a sequence converges almost everywhere. In recent years, the scholars from different countries have made many efforts on the generalization of the lemma and also acquired a series of good results. And most of these results are mainly directed against weakening the condition of the second part of the Borel-Cantelli lemma and give the conditional Borel-Cantelli lemma and weighted Borel-Cantelli lemma. In this paper on the basis of documents available, two Borel-Cantelli lemmas of random variable sequences and a conditional Borel-Cantelli lemma are given, which are the generalization of the existing results. And a conditional version of the law of large numbers is proved using the conditional Kolmogorov inequality and conditional Borel-Cantelli lemma.This paper is divided into three parts. The first part is the introduction, including the basic concepts, inequalities and the related properties. The second part is the Borel-Cantelli lemma of random variables sequences, which is the generalization of the weighted Borel-Cantelli lemma. The third part is conditional Borel-Cantelli lemma of random variable sequences, and a conditional Borel-Cantelli lemma is given, which is applied to prove the conditional law of large numbers.