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

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 , T n = ∑ i = 1 n S 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 { ( 2 k ∏ j = 1 k T j k ! ( k + 1 ) ! 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 variables e 10 / 3 N and N is a standard normal random variable.