Lithuanian Mathematical Journal

, Volume 55, Issue 2, pp 159–192 | Cite as

Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters*



Stoev and Taqqu introduced a linear multifractional stable motion (LMSM), an extension of a linear fractional stable motion (LFSM) such that the Hurst parameter H becomes a function H(t). The stability parameter α determines tail heaviness of marginal distributions of LMSM. Under some conditions, Stoev and Taqqu showed that H(t 0) is its self-similarity exponent at a t 0 ≠ 0; also, recently, Ayache and Hamonier established that H(t 0)−1/α and min tI H(t)−1/α are its local Hölder exponent at t 0 and uniform Hölder exponent on a compact interval I.

We construct, strongly consistent wavelet estimators of min tI H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in \( \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) \).


stable stochastic processes wavelet coefficients Hölder regularity local self-similarity laws of large numbers 


60G22 60G52 60M09 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.UMR CNRS 8524, Laboratoire Paul Painlevé, Bât. M2, Université Lille 1Villeneuve d’Ascq CedexFrance
  2. 2.EA 2694, Laboratoire de Biomathématiques, Université Lille 2Lille CedexFrance

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