Conditional Borel-Cantelli Lemma And Its Applications

Probability theory is a subject of random phenomenon,which has a wide range of applications in physics,economy and communication.And in probability theory,Borel-Cantelli lemma is a very important lemma and plays an important role in the proof of limit theories.Therefore,studying the conditional versions of Borel-Cantelli lemma is of great significance.This paper consists of the following parts:In the first part,we introduce give three representations of the traditional Borel-Cantelli lemma and the background of the study.In the second part,we get some lemma about the number of relations,and then gets the upper and lower limits of the events.In the third part,we introduce the condition Borel-Cantelli lemma according to the upper and lower bounds of the event,which is a generalization of the first and second parts of the traditional Borel-Cantelli lemma.In the fourth part we extend the third expression of the traditional Borel-Cantelli lemma,which shows that in the weaker conditions,the original conclusion holds.And gives a conditional large number theorem,compared with the traditional large number theorem,it relaxes the requirement of independence.The fifth part is also a generalization of the third expression of the traditional Borel-Cantelli lemma,which examines the conditional Borel-Cantelli lemma in the dynamical system.