Precise Asymptotic In The Law Of The Iterated Logarithm Of PA And Martingale Difference Sequences

The random variables that come from practical problems are usually not independent.There are always some dependences or another among random variables. Therefbre, theproperties of the dependent random variables had drawn many attentions from scholars.Positively associated random variables (PA) and Martingale difference sequences arevery important cases in the dependent random variables. Positively associated randomvariables were introduced in the statistical literature by Esary et al. in 1967 and havefound many applications in reliability theory and percolation theory. Martingale differ-ence sequences were the natural extension of independent random variables, which wereintroduced into modern probabilistic literature by Ville in 1939. The concept of martin-gale difference is of important significance in theory and application, for it has a strongvisual background.The law of the iterated logarithm was introduced in the probabilistic literature byKhintchine in 1924; it is the precise phenomenon of the strong law of large numbers.Precise asymptotic is extensions of the weighted series of random variables. Gut et al.have many contributions on this direction. In this chapter, under some suitable conditions,we shall extend the corresponding results of Gut and Spataru (2000a) to the positivelyassociated sequences and the martingale difference sequences.This thesis consists of three chapters:Chapter 1, we simply introduce the background of this thesis and present some im-portant lemmas. Which including: Lemma 1.2.3 we obtainfor small enoughε＞0 uniformly to set up.Lemma 1.2.6 Let b′(ε; M)=b(1/ε; M), we obtainfor big enoughε＞0 uniformly to set up.In the second chapter, we consider the precise asymptotic in the law of the iteratedlogarithm of PA. In the first part we introduce some lemmas about PA, which can be usein the last part. And in the second, third parts we present the conclusions and the processof proof that extending the corresponding results of Gut and Spataru to the positivelyassociated sequences. The conclusions are including:Theorem 2.2.1 Let a_n=O(1/loglogn). For any b＞-1, r=2(b+2) and someδ＞0, satisfying u(n)=O(n-(r-2)(r+δ)/(2δ)), we haveandwhereΓ(·) is Gamma function.Theorem 2.2.2 For any b＞-1, we have In the third chapter, we consider the precise asymptotic in the law of the iterated log-arithm martingale difference sequences. In the first part we introduce some lemmas aboutmartingale difference sequences, which can be use in last part. And in the second, thirdparts we present the conclusions and the process of proof that extending the correspondingresults of Gut and Spataru to the martingale difference sequences. The conclusions areincluding:Theorem 3.2.1 let a_n=O(1/loglogn). For any b＞-1, satisfying (sum from i=n to n(X_i~2))/nσ~2(?)1We haveandwhereΓ(·) is Gamma function.Theorem 3.2.2 For b＞-1, satisfying (sum from i=n to n(X_i~2))/nσ~2(?)1 We have...