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

Abstract

Let { X n , n ⩾ 1 } be a strictly stationary positively or negatively associated sequence of positive random variables with E X 1 = μ > 0 and Var ( X 1 ) = σ 2 < ∞ . Denote S n = ∑ i = 1 n X i and γ = σ / μ the coefficient of variation. Under suitable conditions, we show that ∀ x lim n → ∞ 1 log n ∑ k = 1 n 1 k I ∏ j = 1 k S j k ! μ k 1 / ( γ σ 1 k ) ⩽ x = F ( x ) a . s ., where σ 1 2 = 1 + 2 σ 2 ∑ j = 2 ∞ Cov ( X 1 , X j ) , F ( · ) is the distribution function of the random variable e 2 N , and N 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]).