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

Let {Xn,n⩾1} be a strictly stationary positively or negatively associated sequence of positive random variables with EX1=μ>0EX1=μ>0, and VarX1=σ2<∞VarX1=σ2<∞. Denote Sn=∑i=1nXi, Tn=∑i=1nSi and γ=σ/μγ=σ/μ the coefficient of variation. Under suitable conditions, we show that∀xlimn→∞1logn∑k=1n1kI{(2k∏j=1kTjk!(k+1)!μk)1/(γσ1k)⩽x}=F(x)a.s., where σ12=1+2σ2∑j=2∞Cov(X1,Xj), F(⋅)F(⋅) is the distribution function of the random variables e10/3N and NN is a standard normal random variable.