Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

Werner Linde,Antoine Ayache

doi:10.1214/ejp.v14-727

Abstract

The aim of the present paper is to investigate series representations of the Riemann-Liouville process $R^\alpha$, $\alpha >1/2$, generated by classical orthonormal bases in $L_2[0,1]$. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of $R^\alpha$ via the trigonometric system possesses the optimal convergence rate if and only if $1/2 3/2$ a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases $\alpha > 3/2$ and $\alpha = 3/2$. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case $\alpha = 3/2$ a representation of the Riemann-Liouville process via the Haar system is also not optimal.

Highlights

Let X = (X (t))t∈T be a centered Gaussian process over a compact metric space (T, d) possessing a.s. continuous paths
The aim of the present paper is to investigate those questions for the Riemann–Liouville process Rα defined by
(1.5) is already a series representation and we shall prove in Theorem 4.8 that it is optimal for all α > 1/2

Summary

Introduction

Let X = (X (t))t∈T be a centered Gaussian process over a compact metric space (T, d) possessing a.s. continuous paths. Even in the cases where these representations are not optimal it might be of interest how fast (or slow) the error in (1.2) tends to zero as n → ∞. We have lower and upper estimates which differ by log n Another process, tightly related to Rα, is the Weyl process Iα which is stationary and 1–periodic. (1.5) is already a series representation and we shall prove in Theorem 4.8 that it is optimal for all α > 1/2. If α > 1/2 is an integer, their difference is even a process of finite rank

Approximation by a Fixed Basis

A General Approach

The Trigonometric System

Some useful notations and some preliminary results