Central limit theorem under variance uncertainty

Dmitry Rokhlin

doi:10.1214/ecp.v20-4341

Abstract

We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables $\xi_j$, perturbed by predictable multiplicative factors $\lambda_j$ with values in intervals $[\underline\lambda_j,\overline\lambda_j]$. It is assumed that the sequences $\underline\lambda_j$, $\overline\lambda_j$ are bounded and satisfy some stabilization condition. Under the classical Lindeberg condition we show that the CLT limit, corresponding to a "worst'' sequence $\lambda_j$, is described by the solution $v$ of one-dimensional $G$-heat equation. The main part of the proof follows Peng's approach to the CLT under sublinear expectations, and utilizes Hölder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.

Highlights

Consider a sequence of independent one-dimensional random variables∞ j=1 with zero means and finite variances σj2 = Eξj2 > 0
We prove the central limit theorem (CLT) for a sequence of independent zero-mean random variables ξj, perturbed by predictable multiplicative factors λj with values in intervals [λj, λj ]
Under the classical Lindeberg condition we show that the CLT limit, corresponding to a “worst” sequence λj, is described by the solution v of one-dimensional G-heat equation

Summary

Introduction

Consider the space of sequences Ω = {(yi)∞ i=1 : yi ∈ R}, and introduce the space of random variables H as follows: H = ∪∞ n=1Hn, where Hn is some linear space (we do not go into details) of functions Y = ψ(y1, . Note that if there is no model uncertainty: λj = λj = 1, Theorem 1.4 reduces to the classical CLT, mentioned at the beginning of the present paper. This is not the case with the result of [20], since in this case the conditions (1.12), (1.13) are stronger the Lindeberg condition.

By the Hölder continuity of vt we have

The desired result is a consequence of the relations

Just mentioned inequalities imply that