October 2023 Estimation of mixed fractional stable processes using high-frequency data
Fabian Mies, Mark Podolskij
Author Affiliations +
Ann. Statist. 51(5): 1946-1964 (October 2023). DOI: 10.1214/23-AOS2312


The linear fractional stable motion generalizes two prominent classes of stochastic processes, namely stable Lévy processes, and fractional Brownian motion. For this reason, it may be regarded as a basic building block for continuous time models. We study a stylized model consisting of a superposition of independent linear fractional stable motions and our focus is on parameter estimation of the model. Applying an estimating equations approach, we construct estimators for the whole set of parameters and derive their asymptotic normality in a high-frequency regime. The conditions for consistency turn out to be sharp for two prominent special cases: (i) for Lévy processes, that is, for the estimation of the successive Blumenthal–Getoor indices and (ii) for the mixed fractional Brownian motion introduced by Cheridito. In the remaining cases, our results reveal a delicate interplay between the Hurst parameters and the indices of stability. Our asymptotic theory is based on new limit theorems for multiscale moving average processes.

Funding Statement

Mark Podolskij gratefully acknowledges financial support of ERC Consolidator Grant 815703 “STAMFORD: Statistical Methods for High-Dimensional Diffusions”.


The authors would like to thank Carsten Chong for pointing out an error in a previous version, and two anonymous referees for their very careful reading and constructive comments, which helped to improve the article significantly.

This work was done while Fabian Mies was postdoctoral researcher at the Institute of Statistics, RWTH Aachen University, Germany.


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Fabian Mies. Mark Podolskij. "Estimation of mixed fractional stable processes using high-frequency data." Ann. Statist. 51 (5) 1946 - 1964, October 2023. https://doi.org/10.1214/23-AOS2312


Received: 1 August 2022; Revised: 1 June 2023; Published: October 2023
First available in Project Euclid: 14 December 2023

Digital Object Identifier: 10.1214/23-AOS2312

Primary: 60G22 , 62F12
Secondary: 60F05 , 62E20 , 62M09

Keywords: High frequency data , Lévy processes , linear fractional stable motion , Parametric estimation , Self-similar processes

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 5 • October 2023
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