Asymptotics for sums of a function of normalized independent sums

TL;DR

This paper establishes a central limit theorem for sums of functions of normalized sums of i.i.d. variables, extending previous results to weaker assumptions and broadening the theoretical understanding of such sums.

Contribution

It generalizes a known CLT for functions of normalized sums, demonstrating its validity under weaker conditions than previously established.

Findings

The CLT applies to a wider class of functions.

Weaker assumptions are sufficient for the theorem to hold.

The result extends previous work by Rempała and Wesołowski.

Abstract

We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular we show that previously known result from Rempa\la and Weso\lowski (Statist. Probab. Lett. 74 (2005) 129--138), which can be obtained by applying the logarithm as the function, holds true under weaker assumptions.