A study of the limiting behavior of delayed random sums under non-identical distributions setup

TL;DR

This paper investigates the asymptotic behavior of delayed sums of independent, non-identically distributed random variables, establishing laws of the iterated logarithm under certain conditions, extending previous results.

Contribution

It extends existing laws of the iterated logarithm to delayed sums with non-identical distributions and random delays, broadening understanding of their limiting behavior.

Findings

Established laws of the iterated logarithm for delayed sums.

Extended previous results to non-identical distribution setups.

Provided conditions under which the limiting behavior holds.

Abstract

We consider delayed sums of the type S_{n+an}-Sn where a_n is possibly a positive integer valued random variable satisfying certain conditions and S_n is the sum of independent random variables X_n with distribution functions F_n in {G_1, G_2} . We study the limiting behavior of delayed sums and prove laws of the iterated logarithm of Chover- type. These results extend the results in Vasudeva and Divanji (1992) and Chen (2008).