An almost sure central limit theorem for products of sums under association

Let {Xn,n⩾1}{Xn,n⩾1} be a strictly stationary positively or negatively associated sequence of positive random variables with EX1=μ>0EX1=μ>0 and Var(X1)=σ2<∞Var(X1)=σ2<∞. Denote Sn=∑i=1nXi and γ=σ/μγ=σ/μ the coefficient of variation. Under suitable conditions, we show that∀xlimn→∞1logn∑k=1n1kI∏j=1kSjk!μk1/(γσ1k)⩽x=F(x)a.s.,where σ12=1+2σ2∑j=2∞Cov(X1,Xj), F(·)F(·) is the distribution function of the random variable e2N, and NN is a standard normal random variable. This extends the earlier work on independent, positive random variables (see Khurelbaatar and Rempala [2006. A note on the almost sure limit theorem for the product of partial sums. Appl. Math. Lett. 19, 191–196]).