Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles

The Annals of Statistics · 2008 · Volume 36 · Issue 3

Abstract

This paper is devoted to the introduction of a new class of consistent estimators of the fractal dimension of locally self-similar Gaussian processes. These estimators are based on convex combinations of sample quantiles of discrete variations of a sample path over a discrete grid of the interval [0, 1]. We derive the almost sure convergence and the asymptotic normality for these estimators. The key-ingredient is a Bahadur representation for sample quantiles of nonlinear functions of Gaussian sequences with correlation function decreasing as k−αL(k) for some α>0 and some slowly varying function L(⋅).

Keywords

60G18, 62G30, Bahadur representation of sample quantiles, fractional Brownian motion, Hurst exponent estimation, Locally self-similar Gaussian process

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