An Invariance Principle of G-Brownian Motion for the Law of the Iterated Logarithm under G-expectation

TL;DR

This paper extends the classical law of the iterated logarithm to the G-expectation framework, establishing an invariance principle for G-Brownian motion that broadens its applicability in nonlinear expectation spaces.

Contribution

It introduces a novel invariance principle for G-Brownian motion, generalizing the classical LIL under the nonlinear G-expectation setting.

Findings

Proves the invariance principle of G-Brownian motion for LIL.

Demonstrates the applicability of LIL in nonlinear expectation spaces.

Extends classical probability results to G-expectation framework.

Abstract

The classical law of the iterated logarithm (LIL for short)as fundamental limit theorems in probability theory play an important role in the development of probability theory and its applications. Strassen (1964) extended LIL to large classes of functional random variables, it is well known as the invariance principle for LIL which provide an extremely powerful tool in probability and statistical inference. But recently many phenomena show that the linearity of probability is a limit for applications, for example in finance, statistics. As while a nonlinear expectation--- G-expectation has attracted extensive attentions of mathematicians and economists, more and more people began to study the nature of the G-expectation space. A natural question is: Can the classical invariance principle for LIL be generalized under G-expectation space? This paper gives a positive answer. We present the invariance principle of G-Brownian motion for the law of the iterated logarithm under G-expectation.