The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation

TL;DR

This paper investigates the moments of the maximum normalized partial sums of i.i.d. variables under sub-linear expectations, establishing conditions for finiteness and applying results to laws of the iterated logarithm for moving average processes.

Contribution

It provides necessary and sufficient conditions for the finiteness of moments of maximum normalized partial sums under sub-linear expectations, extending classical LIL results.

Findings

Established conditions for moments to be finite

Derived law of the iterated logarithm under sub-linear expectations

Applied results to moving average processes

Abstract

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.