Strong limit theorems for extended independent and extended negatively dependent random variables under non-linear expectations

TL;DR

This paper establishes strong limit theorems like the law of large numbers and the law of the iterated logarithm for sequences of random variables under non-linear expectations, using a weaker form of independence.

Contribution

It introduces a new concept of extended negative dependence and develops powerful inequalities to prove classical limit theorems in non-linear expectation spaces.

Findings

Established strong law of large numbers under extended independence.

Proved law of the iterated logarithm for extended negatively dependent variables.

Improved existing inequalities for extended negative dependence.

Abstract

Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have raised progressive interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng (2008b). We introduce a concept of extended negative dependence which is an extension of this kind of weak independence and the extended negative independence relative to classical probability appeared in recent literatures. Powerful tools as the moment inequality and Kolmogorov's exponential inequality are established for this kind of extended negatively independent random variables, which improve those of Chen, Chen and Ng(2010) a lot. And the strong law of large numbers and the law of iterated logarithm are obtained by applying these inequalities.