Some inequalities and limit theorems under sublinear expectations

TL;DR

This paper explores inequalities and limit theorems within the framework of sublinear expectations, establishing key results like Doob's inequality, a version of Kolmogrov's law of large numbers, and a strong law of large numbers under specific conditions.

Contribution

It introduces new inequalities and limit theorems under sublinear expectations, extending classical results to this nonlinear expectation setting.

Findings

Proved Doob's inequality for submartingales under sublinear expectations.

Derived a version of Kolmogrov's law of large numbers.

Established a strong law of large numbers with one-order moment condition.

Abstract

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob's inequality for submartingale and Kolmogrov's inequality. By Kolmogrov's inequality, we obtain a special version of Kolmogrov's law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.