Invariance principles for random sums of random variables

2016/10/09 by Gane Samb Lô, Lo, Gane Samb
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1610.02700

openalex publication_date 2016/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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