Abstract. In this paper we present functional random-sum central limit theorems in $L^2[0,1]$ 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 in $L^2[0,1]$. The almost sure random functional limit theorem for the empirical process in $L^2[0,1]$ is presented, too.

AMS Subject Classification (1991): 60F05, 60F15, 60F17; 60G50

Keyword(s): Almost sure central limit theorem, functional random-sum central limit theorem, logarithmic averages, summation methods, Wiener measure, empirical process

Received January 22, 2007, and in revised form October 18, 2007. (Registered under 6459/2009.)