Fractional Brownian sheet and martingale difference random fields

Zhi Wang,Jing Cui

doi:10.1186/s13660-016-1144-7

Abstract

In this paper, we prove a functional central limit theorem for the multidimensional parameter fractional Brownian sheet using martingale difference random fields. The proof is based on the invariance principle for the Brownian sheet due Poghosyan and Roelly (Stat. Probab. Lett. 38:235-245, 1998).

Highlights

Self-similar stochastic processes with long range dependence are important aspect of stochastic models in various scientific areas, such as econometrics, network traffic analysis, hydrology, telecommunications, and so on
Fractional Brownian motion is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data
It is a suitable generalization of the standard Brownian motion B, which exhibits a long range dependence, self-similarity, and Hölder’s continuity, and which has stationary increments

Summary

Introduction

Self-similar stochastic processes with long range dependence (or long memory) are important aspect of stochastic models in various scientific areas, such as econometrics, network traffic analysis, hydrology, telecommunications, and so on These are processes X = {Xt, t ≥ } whose dependence on the time parameter t is self-similar, in the sense that there exists a (self-similarity) parameter < H < such that for any constant c ≥ , {Xct, t ≥ } and {cH Xt, t ≥ } have the same distribution. Fractional Brownian motion (fBm) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data It is a suitable generalization of the standard Brownian motion B, which exhibits a long range dependence (when H > / ), self-similarity, and Hölder’s continuity, and which has stationary increments.

E Bαt Bαs d

Preliminaries and main results