Marcinkiewicz's strong law of large numbers for non-additive expectation

TL;DR

This paper extends Marcinkiewicz's strong law of large numbers to sub-linear expectation spaces, which model uncertainty with non-additive expectations, using subsequence methods.

Contribution

It introduces a proof of Marcinkiewicz's strong law under non-additive expectations, broadening classical probability results to nonlinear expectation frameworks.

Findings

Established strong law of large numbers in sub-linear expectation space.

Proved convergence of a random series under non-additive expectation.

Extended classical probability theorems to nonlinear expectation models.

Abstract

The sub-linear expectation space is a nonlinear expectation space having advantages of modelling the uncertainty of probability and distribution. In the sub-linear expectation space, we use capacity and sub-linear expectation to replace probability and expectation of classical probability theory. In this paper, the method of selecting subsequence is used to prove Marcinkiewicz type strong law of large numbers under sub-linear expectation space. This result is a natural extension of the classical Marcinkiewicz's strong law of large numbers to the case where the expectation is nonadditive. In addition, this paper also gives a theorem about convergence of a random series.