Harmonizable Multifractional Stable Motion: estimators for the Hurst function and the stability parameter from discrete-time observations, their rates of convergence and the associated central limit theorems
Abstract
Multifractional processes are Gaussian or non-Gaussian extensions of classical Fractional Brownian Motion (FBM) and related fractional processes. Their main feature is that they have a varying Hurst parameter called the Hurst function. In this article, we focus on the Harmonizable Multifractional Stable Motion (HMSM) which extends the non-ergodic Harmonizable Fractional Stable Motion (HFSM). Despite the fact that HMSM was introduced about 15 years ago, no statistical estimator for either of its two parameters, the Hurst function H(•) and the stability parameter α, has been proposed so far in the literature. In our article, it is assumed that the values of a realization of HMSM are observed on the finite subset of the interval (0, 1) formed by the dyadic numbers of a certain even level large enough. A first goal is to construct consistent and asymptotically normal estimators for α -1 := 1/α and the vector H(t1), . . . , H(tq) , where the positive integer q and the distinct points t1, . . . , tq of (0, 1) are arbitrary; and also to provide an estimate of the rate of convergence of these estimators. A second goal is to build, for each compact interval I ⊂ (0, 1), an estimator of the whole Hurst function H(•) on I, which converges to it in probability (and even almost surely under some condition) for the uniform norm on I, and to obtain an estimate of the uniform rate of convergence.
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BibTeX
@misc{ayache2026harmonizable,
title = {{Harmonizable Multifractional Stable Motion: estimators for the Hurst function and the stability parameter from discrete-time observations, their rates of convergence and the associated central limit theorems}},
author = {Antoine Ayache and Christophe Louckx},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05739884v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Harmonizable Multifractional Stable Motion: estimators for the Hurst function and the stability parameter from discrete-time observations, their rates of convergence and the associated central limit theorems
AU - Antoine Ayache
AU - Christophe Louckx
PY - 2026
DA - 2026/09/06
PB - HAL
UR - https://hal.science/hal-05739884v1
AB - Multifractional processes are Gaussian or non-Gaussian extensions of classical Fractional Brownian Motion (FBM) and related fractional processes. Their main feature is that they have a varying Hurst parameter called the Hurst function. In this article, we focus on the Harmonizable Multifractional Stable Motion (HMSM) which extends the non-ergodic Harmonizable Fractional Stable Motion (HFSM). Despite the fact that HMSM was introduced about 15 years ago, no statistical estimator for either of its two parameters, the Hurst function H(•) and the stability parameter α, has been proposed so far in the literature. In our article, it is assumed that the values of a realization of HMSM are observed on the finite subset of the interval (0, 1) formed by the dyadic numbers of a certain even level large enough. A first goal is to construct consistent and asymptotically normal estimators for α -1 := 1/α and the vector H(t1), . . . , H(tq) , where the positive integer q and the distinct points t1, . . . , tq of (0, 1) are arbitrary; and also to provide an estimate of the rate of convergence of these estimators. A second goal is to build, for each compact interval I ⊂ (0, 1), an estimator of the whole Hurst function H(•) on I, which converges to it in probability (and even almost surely under some condition) for the uniform norm on I, and to obtain an estimate of the uniform rate of convergence.
ER -