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On the random functional central limit theorems with almost sure convergence for subsequences

Rychlik Z., Szuster K. S.

Demonstratio Mathematica

2012

Vol. 45, nr 2

283-296

In this paper we present functional random sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related 'logarithmic 'limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random sum central limit theorems for subsequences.

On the random functional central limit theorems with almost sure convergence

Orzóg M., Rychlik Z.

Probability and Mathematical Statistics

2007

Vol. 27, Fasc. 1

125--138

In this paper we present functional random-sum central limit theorems with almost sure convergence for independent nonidentically distributed random variables. We consider the case where the summation random indices and partial sums are independent. In the past decade several authors have investigated the almost sure functional central limit theorems and related ‘logarithmic’ limit theorems for partial sums of independent random variables. We extend this theory to almost sure versions of the functional random-sum central limit theorems.

Some remarks on the almost sure central limit theorem for independent random variables

Rychlik Z., Szuster K. S.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 2

241--249

The purpose of this paper is the proof of an almost sure central limit theorem for subsequences. We obtain an almost sure convergence limit theorem for independent nonidentically distributed random variables. The presented results extend,to nonidentically distributed random variables, theorems given by Schatte [13].