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

2013/02/07 by Antoine Ayache, Ayache, Antoine, Julien Hamonier +1 · 1 citation
Computer Science · Engineering · #Control Systems and Identification #FOS: Mathematics #Fault Detection and Control Systems #Image and Signal Denoising Methods #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1302.1674

openalex publication_date 2013/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Linear fractional stable motion, denoted by \X_H,\al(t)\_{t∈ \R}, is one of the most classical stable processes; it depends on two parameters H∈ (0,1) and \al∈ (0,2). The parameter H characterizes the self-similarity property of \X_H,\al(t)\_{t∈ \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∈ \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.

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• Prediction of linear fractional stable motions using codifference, with application to non-Gaussian rough volatility

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