Central limit theorem under variance uncertainty

TL;DR

This paper establishes a central limit theorem for sequences of independent zero-mean variables with multiplicative variance uncertainty, linking the limit to solutions of a $G$-heat equation under certain conditions.

Contribution

It extends the CLT to cases with variance uncertainty using $G$-expectation and viscosity solutions, providing a new approach to limit characterization.

Findings

CLT holds under variance uncertainty with predictable multiplicative factors.

The limit distribution is characterized by the solution of a $G$-heat equation.

The proof employs Peng's approach and viscosity solution techniques.

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 H\"{o}lder regularity properties of $v$. Under the lack of such properties, we use the technique of half-relaxed limits from the theory of viscosity solutions.