Central limit theorem under uncertain linear transformations

TL;DR

This paper extends the classical central limit theorem to scenarios with uncertain linear transformations, using viscosity solutions of nonlinear PDEs to describe the limit distribution under model uncertainty.

Contribution

It introduces a CLT variant for i.i.d. variables affected by stochastic linear transformations, connecting the limit to the viscosity solution of a G-heat equation.

Findings

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

The proof employs half-relaxed limits from nonlinear PDE approximation theory.

The approach generalizes classical CLT to uncertain linear transformation models.

Abstract

We prove a variant of the central limit theorem (CLT) for a sequence of i.i.d. random variables $\xi_j$, perturbed by a stochastic sequence of linear transformations $A_j$, representing the model uncertainty. The limit, corresponding to a "worst" sequence $A_j$, is expressed in terms of the viscosity solution of the $G$-heat equation. In the context of the CLT under sublinear expectations this nonlinear parabolic equation appeared previously in the papers of S.Peng. Our proof is based on the technique of half-relaxed limits from the theory of approximation schemes for fully nonlinear partial differential equations.