PA序列Chung型对数律的极限定理

Limit theorems of Chung-type law of the logarithm for PA sequences

设{Xn,n≥1}是一均值为零、方差有限的正相伴平稳序列.记Sn=sum Xk,Mn=maxx≤n|Sk|,n≥1 from k=1 to n,并假设0<σ2=EX12+2 sum E X1 Xk<∞ from k=2 to ∞.在E|X1|2+δ<∞,δ∈(0,1],以及对某个α>1,sum Cov(X1,Xj)=O(n-α) from j=n+1 to ∞的条件下,建立了PA序列关于Chung型对数律的精确收敛速度.

Let{Xn;n≥1）be a strictly stationary sequence of positively associated（PA） random variables with mean n zero and finite variance.Denote Sn=sum Xk,Mn=maxx≤n｜Sk｜,n≥1 from k=1 to n,and assume that 0σ2=EX12＋2 sum E X1 Xk∞ from k=2 to ∞.Under the conditions of E｜ X1｜2＋δ∞,whereδ∈（0,1],and sum Cov（X1,Xj）=O（n-α） from j=n＋1 to ∞ someα1,the exact convergence rates of the Chung-type law of the logarithm for PA sequences are obtained.