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

Let {Xn,n[greater-or-equal, slanted]1} be a strictly stationary positively or negatively associated sequence of positive random variables with and . Denote and [gamma]=[sigma]/[mu] the coefficient of variation. Under suitable conditions, we show thatwhere , F(·) is the distribution function of the random variable , and 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]).