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.
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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].
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