Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk=..i=...i+k.i, where {. i ;-. < i < .} is a doubly infinite sequence of identically distributed .-mixing or negatively associated random variables with mean zeros and finite variances, {. i ;-. < i < .} is an absolutely summable sequence of real numbers. Set Sn=.nk=1Xk,n.1. Suppose that .2=E.21+2..k=2E.1.k>0. We prove that for any ..0,ifE[.21(loglog|.1|)..1]<., lim..o.2.+2.n=1.(loglogn).nlognP{|Sn|. ..2nloglogn...........}=1(.+1).....(.+3/2), and if E[.21(log|.1|)..1]<., lim..o.2.+2.n=1.(logn).nP{|Sn|...nlogn.......}= .(2.+2).+1.2.+2, where .=....i=...i,.(.) is a Gamma function and .(2.+2) stands for the (2. + 2)-th absolute moment of the standard normal distribution.