PRECISE RATE IN THE LAW OF ITERATED LOGARITHM FOR ρ-MIXING SEQUENCE

摘要

Let {X,Xn；n≥1}be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance,Set Sn=∑nk=1 Xk,Mn=max k≤n|Sk|,n≥1.Suppose lim n→∞ES2n=t α2>0 and ∞∑n=1ρ2/d(2n)<∞,where d=2,if-12(b+1),if b≥0.It is proved that,for any b>-1,limε2(b+1) ε→0∞∑n=1(log logn)blognP{Mn≥εσ√2nlog logn}=2/(b+1)√πГ(b+3/2)∞∑k=0(-1)k/(2k+1)2b+2,where Г(·)is a Gamma function.