An approximation to the subfractional Brownian sheet using martingale differences

Jinhong Zhang,Guangjun Shen,Mengyu Li

doi:10.1186/s13660-015-0625-4

Abstract

In this paper, we obtain an approximation in law of the subfractional Brownian sheet using martingale differences.

Highlights

As an extension of Brownian motion, Bojdecki et al [ ] introduced and studied the subfractional Brownian motion, a class of self-similar Gaussian processes preserving many properties of the fractional Brownian motion
For H = /, SH coincides with the standard Brownian motion W
Motivated by the aforementioned works, as a first attempt, in this paper, we will prove weak convergence to the subfractional Brownian sheet result for processes construed from the martingale differences sequence in the Skorohord space

Summary

Introduction

As an extension of Brownian motion, Bojdecki et al [ ] introduced and studied the subfractional Brownian motion, a class of self-similar Gaussian processes preserving many properties of the fractional Brownian motion (self-similarity, long-range dependence, Hölder paths). Harnett and Nualart [ ] proved a weak convergence of the Stratonovich integral with respect to a class of Gaussian processes which includes subfractional. We will consider the second one in the two-parameter case and call it a subfractional Brownian sheet This is a centered Gaussian process on R. Motivated by the aforementioned works, as a first attempt, in this paper, we will prove weak convergence to the subfractional Brownian sheet result for processes construed from the martingale differences sequence in the Skorohord space. It can be seen as an extension of the previous results of Shen and Yan [ ] to the two-parameter case. In Section , we prove the main weak convergence Theorem . using the criterion given by Bickel and Wichura [ ] to check the tightness of the approximated processes and the convergence of the finite-dimensional distributions

Preliminaries

Main results