An Almost Sure Central Limit Theorem Of Weighted Sums And Products Of Partial Sums For ?~--Mixing Sequences

Limit theory is an important part in the theory of probability and almost sure central lim-it theorem has been the central topic in the probability theory research,a lot of linear statistics about random samples can be regarded as random variables with the form of weighted sums,therefore study the weighted sums of limit theory is an important part in the application of probability and statistics.Almost sure central limit theorem discuss the distribution of the sums of random vari-ables is normal distribution in probability theory and a set of theorems is the theoretical basis of mathematical statistics and error analysis,pointed out that under certain conditions,the distribution of the sums of a large number of random variables can be approximately considered as the normal distribution.This text research the almost sure central limit theorem of weighted sums and products of partial sums for ?~--mixing sequences based on the foundation of the almost sure central limit theorem for weighted sums under NA sequences and products of partial sums for ?~--mixing sequences.Then comes to the following conclusions:Theorem1 Let {Xn,n ?1} be a strictly stationary sequence of ?~--mixing with zero mean,and 0<E|X1|r<?,as r>2,let {ani,1 ?i?n,n ?1} be an triabgle array of real n nonnegative numbers,and Sn=(?)aniXi.Assume that(1)sup(?)ani2<?,|ani|?Cn1/2/1?log?(n/i),1?i?n,,n?1,(?)?>0;(2)(?)Cov(X1,Xj)|<?;(3)?-(n)= O(log-?n),?>1;(4)Var(Sn)=1,as n??.Then,for every x?R(?)dkI{Sk?x}=?(x).(1)where,Dn=(?)dk,dk = exp(log?k)/k,??[0,1/2).Theorem 2 As =(?)dk,dk=olgrk/k,r>-1,and meet the conditions of theorem L,the conclusion of theoreml still true.Theorem 3 Let {Xn,n ?1} be a strictly stationary ?'-mixing sequence of positive random variables,with E|X1|=?>0,VarX1= ?2<?,and0<E|X1|r<?,as r>2,let Sn=(?)Xi,and ?=?/? the coefficient of variation,Assume that(1)0<?12=EX12+2(?)Cov(X1,Xj)<?;(2)(?)|Cov(X1,Xj)|<?;(3)?-(n)= O(log-?n),?>1;(4)inf n?N?2n/n>0.Then,for every x?R,(?)(2)Where F(x)is the distribution fiunction of the randoin variable e(?),and N is a standard nor-mal random variable.Dn=(?)dk,dk=exp(lognk)/k,??[0,1/2),?1n=Var(Sn,n),Sn,n=(?)bk,nYk,Yk=Xk-?/?,k?1,bk,n=(?)1/i,k?n,bk,n=0,k>n.Theorem 4 As Dn=(?)dk,dk=logrk/k,r>-1,and meet the conditions of theorem 3,the conclusion of theorem3 still true.