Convergence in Total Variation of Random Sums

Luca Pratelli,Pietro Rigo

doi:10.3390/math9020194

Abstract

Let (Xn) be a sequence of real random variables, (Tn) a sequence of random indices, and (τn) a sequence of constants such that τn→∞. The asymptotic behavior of Ln=(1/τn)∑i=1TnXi, as n→∞, is investigated when (Xn) is exchangeable and independent of (Tn). We give conditions for Mn=τn(Ln−L)⟶M in distribution, where L and M are suitable random variables. Moreover, when (Xn) is i.i.d., we find constants an and bn such that supA∈B(R)|P(Ln∈A)−P(L∈A)|≤an and supA∈B(R)|P(Mn∈A)−P(M∈A)|≤bn for every n. In particular, Ln→L or Mn→M in total variation distance provided an→0 or bn→0, as it happens in some situations.

Highlights

All random elements appearing in this paper are defined on the same probability space, say (Ω, A, P).A random sum is a quantity such as ∑iT=n1 Xi, where (Xn : n ≥ 1) is a sequence of real random variables and (Tn : n ≥ 1) a sequence of N-valued random indices
We prove a weak law of large numbers (WLLN), a central limit theorem (CLT), and we investigate the rate of convergence with respect to the total variation distance
Ln → L in total variation distance provided the conditions of Theorem 4 hold, a = 0, L(V) is absolutely continuous, and lim n τn E

Summary

Introduction

All random elements appearing in this paper are defined on the same probability space, say (Ω, A, P). Under such conditions, we prove a weak law of large numbers (WLLN), a central limit theorem (CLT), and we investigate the rate of convergence with respect to the total variation distance. The total variation distance between two probability measures on B(S), say μ and ν, is dTV (μ, ν) = sup |μ(A) − ν(A)|. With a slight abuse of notation, if X and Y are S-valued random variables, we write dTV (X, Y) instead of dTV L(X), L(Y) , namely dTV (X, Y) = sup |P(X ∈ A) − P(Y ∈ A)|.

Stable Convergence

Rate of Convergence with Respect to Total Variation Distance