Almost Sure Central Limit Theorem in Sub-linear Expectation Spaces

TL;DR

This paper extends the classical almost sure central limit theorem to sub-linear expectation spaces, providing a quasi sure convergence version and a strong law of large numbers for non-additive probabilities.

Contribution

It introduces a new almost sure central limit theorem under sub-linear expectations and proves a strong law of large numbers without independence or identical distribution assumptions.

Findings

Establishment of a quasi sure convergence version of the central limit theorem.

Extension of classical results to non-additive probability frameworks.

Proof of a strong law of large numbers under weaker conditions.

Abstract

Peng (2006) initiated a new kind of central limit theorem under sub-linear expectations. Song (2017) gave an estimate of the rate of convergence of Peng's central limit theorem. Based on these results, we establish a new kind of almost sure central limit theorem under sub-linear expectations in this paper, which is a quasi sure convergence version of Peng's central limit theorem. Moreover, this result is a natural extension of the classical almost sure central limit theorem to the case where the probability is no longer additive. Meanwhile, we prove a new kind of strong law of large numbers for non-additive probabilities without the independent identically distributed assumption.