Linear fractional stable motion: a wavelet estimator of the $\al$ parameter

TL;DR

This paper introduces a wavelet-based estimator for the tail parameter  of linear fractional stable motion, demonstrating its strong consistency and asymptotic properties based on wavelet coefficient analysis.

Contribution

The paper develops a novel wavelet estimator for  in linear fractional stable motion, providing theoretical proof of its strong consistency under known H.

Findings

Wavelet coefficients' maxima scale as 2^{-j(H-1/)}

The estimator for  is strongly consistent

Asymptotic behavior of wavelet coefficients is characterized

Abstract

Linear fractional stable motion, denoted by $\{X_{H,\al}(t)\}_{t\in \R}$, is one of the most classical stable processes; it depends on two parameters $H\in (0,1)$ and $\al\in (0,2)$. The parameter $H$ characterizes the self-similarity property of $\{X_{H,\al}(t)\}_{t\in \R}$ while the parameter $\al$ governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that $H>1/\al$ and that $H$ is known. We show that, on the interval $[0,1]$, the asymptotic behaviour of the maximum, at a given scale $j$, of absolute values of the wavelet coefficients of $\{X_{H,\al}(t)\}_{t\in \R}$, is of the same order as $2^{-j(H-1/\al)}$; then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter $\al$.