Almost sure convergence for self-normalized products of sums of partial sums of ρ − mixing sequences


Qunying Wu, Yuanying Jiang




Let X,X1,X2, . . . be a stationary sequence of ρ−-mixing positive random variables. A universal result in the area of almost sure central limit theorems for the self-normalized products of sums of partial sums ( ∏k j=1(T j/( j( j + 1)µ/2)))µ/(βVk) is established, where: T j = ∑ j i=1 Si,Si = ∑i k=1 Xk,Vk = √∑k i=1 X2i , µ = EX, β > 0. Our results generalize and improve those on almost sure central limit theorems obtained by previous authors from the independent case to ρ−-mixing sequences and from partial sums case to self-normalized products of sums of partial sums