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

Let X,X1,X2,... be a stationary sequence of ??-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 (?kj =1(Tj/(j(j+1)?/2)))?=(?Vk) is established, where: Tj = ?ji=1 Si,Si = ?i k=1 Xk,Vk = ??ki=1 X2i,? = EX, ? > 0. Our results generalize and improve those on almost sure central limit theorems obtained by previous authors from the independent case to ??-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.

Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables

We study the weak convergence in the spaceD[0,1]of processes constructed from products of sums of independent but not necessarily identically distributed random variables. The presented results extend and generalize limit theorems known so far for i.i.d. sequences.

Weak and Almost Sure Convergence for Products of Sums of Associated Random Variables

We study weak convergence of product of sums of stationary sequences of associated random variables to the log-normal law. The almost sure version of this result is also presented. The obtained theorems extend and generalize some of the results known so far for independent or associated random variables.