Pseudo-independence, independence and related limit theorems under sublinear expectations

TL;DR

This paper defines pseudo-independence within sublinear expectation spaces, explores its relation to Peng's independence, and establishes law of large numbers and central limit theorems for pseudo-independent variables, including counterexamples highlighting the importance of moment conditions.

Contribution

It introduces pseudo-independence in sublinear expectations and proves related limit theorems, providing new insights and a probabilistic proof of Peng's law of large numbers.

Findings

Pseudo-independence is defined via classical conditional expectation.

Law of large numbers and CLT are established for pseudo-independent variables.

Counterexamples show the necessity of moment conditions for Peng's LLN and CLT.

Abstract

This paper introduces the notion of pseudo-independence on the sublinear expectation space $(\Omega,\mathcal{F},\mathcal{P})$ via the classical conditional expectation, and the relations between pseudo-independence and Peng's independence are detailed discussed. Law of large numbers and central limit theorem for pseudo-independent random variables are obtained, and a purely probabilistic proof of Peng's law of large numbers is also given. In the end, some relevant counterexamples are indicated that Peng's law of large numbers and central limit theorem are invalid only with the first and second moment conditions respectively.