On the functional limits for sums of a function of partial sums

TL;DR

This paper establishes a functional central limit theorem for normalized sums of functions of partial sums of i.i.d. variables, extending previous results on products of partial sums using a novel technique.

Contribution

It generalizes existing limit theorems from products to functions of partial sums via a new methodological approach.

Findings

Proves a new functional CLT for functions of partial sums.

Extends previous results from products to more general functions.

Uses a technique from Huang and Zhang (2007) for the proof.

Abstract

We derive a functional central limit theorem (fclt) for normalised sums of a function of the partial sums of independent and identically distributed random variables. In particular, we show, using a technique presented in Huang and Zhang (Electron. Comm. Probab. 12 (2007), 51--56), that the result from Qi (Statist. Probab. Lett. 62 (2003), 93--100), for normalised products of partial sums, can be generalised in this fashion to a fclt.