Almost sure local central limit theorem for the product of some partial sums with optimized weight. Issue 13 (3rd July 2021) - Xu, Feng; Wang, Binhui; Hou, Yawen
Abstract: The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let X k, k ≥ 1 be a sequence of independent and identically distributed (i.i.d.) positive random variables. Under a fairly general condition an universal result in almost sure local limit theorem for the product of some partial sums ( ∏ i = 1 k S k, i / ( ( k − 1 ) k μ k ) ) 1 / ( γ k ) is established on the weight d k = k − 1 exp ( log β k ), 0 ≤ β < 1 / 2, where S k, i = ∑ j = 1 k X j − X i, E X 1 = μ and the coefficient of variation γ = σ / μ, where Var X 1 = σ 2 .