Invariance principles for random sums of random variables

TL;DR

This paper establishes invariance principles for sums of random variables indexed by a stochastic process, demonstrating their convergence to Gaussian processes and Brownian motion under certain conditions.

Contribution

It introduces new invariance principles for random sums with stochastic indexing, extending classical results to more general stochastic processes.

Findings

Sums of random variables converge to Gaussian processes under specific conditions.

Rescaling based on the expectation of the stochastic process yields convergence to Brownian motion.

Provides sufficient conditions for weak convergence of these sums to continuous Gaussian limits.

Abstract

This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.