The almost sure local central limit theorem for products of partial sums under negative association.

Let { X n , n ≥ 1 } be a strictly stationary 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 , p k = P ( a k ≤ ( ∏ j = 1 k S j / ( k ! μ k ) ) 1 / ( γ σ 1 k ) < b k ) and γ = σ / μ the coefficient of variation. Under some suitable conditions, we derive the almost sure local central limit theorem lim n → ∞ 1 log n ∑ k = 1 n 1 k p k I { a k ≤ ( ∏ j = 1 k S j k ! μ k ) 1 / ( γ σ 1 k ) < b k } = 1 a.s., where σ 1 2 = 1 + 1 σ 2 ∑ j = 2 ∞ Cov ( X 1 , X j ) > 0 .