An Extension of the Almost Sure Central Limit Theorem for Products of Sums Under Association
We show that if {X n , n ≥ 1} is a strictly stationary positively or negatively associated sequence of positive random variables with , and (d k ) is a positive numerical sequence obeying a condition similar to Kolmogorov's condition for the LIL, then letting we have where , γ = σ/μ the coefficient of variation, F(·) is the distribution function of the random variable , and 𝒩 is a standard normal random variable. This shows that logarithmic means, used traditionally in a.s. central limit theory for products of sums, can be replaced by other means, leading to considerably sharper results.