Convergence to a self-normalized G-Brownian motion

Zhengyan Lin,Li-Xin Zhang

doi:10.1186/s41546-017-0013-8

Zhengyan Lin, Li-Xin Zhang

Open Access

https://doi.org/10.1186/s41546-017-0013-8

Copy DOI

Abstract

G-Brownian motion has a very rich and interesting new structure that nontrivially generalizes the classical Brownian motion. Its quadratic variation process is also a continuous process with independent and stationary increments. We prove a self-normalized functional central limit theorem for independent and identically distributed random variables under the sub-linear expectation with the limit process being a G-Brownian motion self-normalized by its quadratic variation. To prove the self-normalized central limit theorem, we also establish a new Donsker’s invariance principle with the limit process being a generalized G-Brownian motion.

Highlights

Let {Xn; n ≥ 1} be a sequence of independent and identically distributed random variables on a probability space (, F, P )
The purpose of this paper is to establish the self-normalized central limit theorem under the sub-linear expectation
The central limit theorem under the sub-linear expectation was first established by

Summary

Introduction

Keywords Sub-linear expectation · G-Brownian motion · Central limit theorem · Invariance principle · Self-normalization Let {Xn; n ≥ 1} be a sequence of independent and identically distributed random variables on a probability space ( , F , P ). The purpose of this paper is to establish the self-normalized central limit theorem under the sub-linear expectation. The central limit theorem under the sub-linear expectation was first established by

Objectives

Results

Conclusion