Linear Multifractional Stable Motion: representation via Haar basis

TL;DR

This paper presents a new wavelet series representation of Linear Multifractional Stable Motion using Haar wavelets, providing explicit formulas, convergence results, and a novel simulation method for the process.

Contribution

It introduces a Haar wavelet-based series representation of LMSM, improving explicitness over previous methods and enabling new simulation techniques.

Findings

Series converge almost surely in continuous function space

Explicit rates of convergence are established

New method for simulating LMSM's frequency components

Abstract

The aim of this paper is to give a wavelet series representation of Linear Multifractional Stable Motion (LMSM in brief), which is more explicit than that introduced in (Ayache & Hamonier 2012). Instead of using Daubechies wavelet, which is not given by a closed form, we use the Haar wavelet. In order to obtain this new representation, we introduce a Haar expansion of the high and low frequency parts of the $\mathcal{S}\alpha \mathcal{S}$ random field $X$ generating LMSM. Then, by using Abel transforms, we show that these series are convergent, almost surely, in the space of continuous functions. Finally, we determine their almost sure rates of convergence in the latter space. Note that these representations of the high and low frequency parts of $X$, provide a new method for simulating the high and low frequency parts of LMSM.