Functional central limit theorem with mean-uncertainty under sublinear expectation

TL;DR

This paper develops a new functional central limit theorem under mean-uncertainty within sublinear expectation spaces, extending classical results to uncertain mean scenarios with probabilistic proofs.

Contribution

It introduces a novel CLT with mean-uncertainty on the canonical space and extends it to general sublinear expectation spaces, generalizing Peng's CLT.

Findings

Established a new functional CLT with mean-uncertainty

Extended the CLT to sublinear expectation spaces

Applied results to the two-armed bandit problem

Abstract

In this paper, we introduce a fundamental model for independent and identically distributed sequence with model uncertainty on the canonical space $(\mathbb{R}^\mathbb{N},\mathcal{B}(\mathbb{R}^\mathbb{N}))$ via probability kernels. Thanks to the well-defined upper and lower variances, we obtain a new functional central limit theorem with mean-uncertainty on the canonical space by the method based on the martingale central limit theorem and stability of stochastic integral in the classical probability theory. Then we extend it to the general sublinear expectation space through a new representation theorem. Our results generalize Peng's central limit theorem with zero-mean to the case of mean-uncertainty and provides a purely probabilistic proof instead of the existing nonlinear partial differential equation approach. As an application, we consider the two-armed bandit problem and generalize the corresponding central limit theorem from the case of mean-certainty to mean-uncertainty.