The Study Of Borel-Cantelli Lemma And The Necessary And Sufficient Conditions To The Strong Law Of Large Numbers

As we all know,the theory of probability limit plays a cornerstone role in probability theory and mathematical statistics,and the research on the Borel-Cantelli lemma and strong law of large numbers is an important research con-tent.Therefore,in this thesis we will mainly study these two problems.First of all,many scholars have studied the related problems of Borel-Cantelli lemma in the probability space.For example,for the sequence of independent event or a dependent event,some upper and lower bounds of Borel-Cantelli lem-ma were obtained under some conditions.However,there is little research on the Borel-Cantelli lemma in the general measurement space.We will study the Borel-Cantelli lemma for the non-negative functions in a general finite measure space,and obtain some upper and lower bounds which generalize some classi-cal results.Furthermore,some examples including dependent random variables are illustrated to our results.Secondly,for the study of the law of large num-bers,many scholars have extended the results for the independent case to the dependent case.But most of them only gave the sufficient conditions to the strong law of large numbers.So the research on its necessary conditions should be studied.We will study the necessary and sufficient conditions to the strong law of large numbers under the dependent sequence-NSD sequence.Under some functional conditions,we obtain the necessary and sufficient conditions to the strong law of large numbers with NSD sequences.As an application,it can be immediately obtained the Marcinkiwicz type and logarithmic mean type strong laws of large numbers for NSD sequences.Finally,with the help of MATLAB software,we do some simulations to verify our laws of large numbers we obtained.