On the functional central limit theorem with mean-uncertainty

TL;DR

This paper develops a new functional central limit theorem for sequences with mean-uncertainty using probabilistic methods, extending classical results to sublinear expectation spaces without PDE reliance.

Contribution

It introduces a novel model for i.i.d. sequences under model uncertainty and proves a functional CLT in this setting, extending classical probability theory.

Findings

Establishes a functional CLT with mean-uncertainty for i.i.d. sequences.

Extends the CLT to sublinear expectation spaces.

Provides purely probabilistic proofs without PDE methods.

Abstract

We introduce a new basic model for independent and identical distributed sequence on the canonical space $(\mathbb{R}^\mathbb{N},\mathcal{B}(\mathbb{R}^\mathbb{N}))$ via probability kernels with model uncertainty. Thanks to the well-defined upper and lower variances, we obtain a new functional central limit theorem with mean-uncertainty by the means of martingale central limit theorem and stability of stochastic integral in the classical probability theory. Then we extend it from the canonical space to the general sublinear expectation space. The corresponding proofs are purely probabilistic and do not rely on the nonlinear partial differential equation.