Linear Multifractional Stable Motion: wavelet estimation of $H(\cdot)$ and $\al$ parameters

TL;DR

This paper develops wavelet-based statistical estimators for the time-varying Hurst function and stability parameter of Linear Multifractional Stable Motion, extending previous models to better capture non-stationary, heavy-tailed processes.

Contribution

It introduces strongly consistent wavelet estimators for the local Hurst function and stability parameter of LMSM, applicable when the Hurst function is sufficiently smooth.

Findings

Estimates of $ ext{min}_{t ext{ in } I} H(t)$, $H(t_0)$, and $eta$ are almost surely consistent.

The estimators work for $eta$ in (1,2) and smooth $H(ullet)$ in a compact interval.

The methodology improves understanding of non-stationary, heavy-tailed stochastic processes.

Abstract

Linear Fractional Stable Motion (LFSM) of Hurst parameter $H$ and of stability parameter $\al$, is one of the most classical extensions of the well-known Gaussian Fractional Brownian Motion (FBM), to the setting of heavy-tailed stable distributions \cite{SamTaq,EmMa}. In order to overcome some limitations of its areas of application, coming from stationarity of its increments as well as constancy over time of its self-similarity exponent, Stoev and Taqqu introduced in \cite{stoev2004stochastic} an extension of LFSM, called Linear Multifractional Stable Motion (LMSM), in which the Hurst parameter becomes a function $H(\cdot)$ depending on the time variable $t$. Similarly to LFSM, the tail heaviness of the marginal distributions of LMSM is determined by $\al$; also, under some conditions, its self-similarity is governed by $H(\cdot)$ and its path roughness is closely related to $H(\cdot)-1/\al$. Namely, it was shown in \cite{stoev2004stochastic} that $H(t_0)$ is the self-similarity exponent of LMSM at a time $t_0\neq 0$; moreover, very recently, it was established in \cite{hamonier2012lmsm}, that the quantities $\min_{t\in I} H(t)-1/\al$, and $H(t_0)-1/\al$, are respectively the uniform H\"older exponent of LMSM on a compact interval $I$, and its local H\"older exponent at $t_0$. The main goal of our article, is to construct, using wavelet coefficients of LMSM, strongly consistent (i.e. almost surely convergent) statistical estimators of $\min_{t\in I} H(t)$, $H(t_0)$, and $\al$; our estimation results, are obtained when $\al\in (1,2)$, and, $H(\cdot)$ is a H\"older function smooth enough, with values in a compact subinterval $[\underline{H},\bar{H}]$ of $(1/\al,1)$.