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

Abstract

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 t∈I 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 t∈I H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in $$ \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) $$ .