Electronic Journal of Statistics

Estimation of the Hurst and the stability indices of a $H$-self-similar stable process

Thi To Nhu Dang and Jacques Istas

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Abstract

In this paper we estimate both the Hurst and the stability indices of a $H$-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $\beta$-negative power variations with $-\frac{1}{2}<\beta<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka’s process, Lévy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and Lévy motion.

Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 4103-4150.

Dates
Received: April 2017
First available in Project Euclid: 24 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1508810900

Digital Object Identifier
doi:10.1214/17-EJS1357

Mathematical Reviews number (MathSciNet)
MR3715823

Zentralblatt MATH identifier
06805088

Keywords
H-sssi processes stable processes self-similarity parameter estimator stability parameter estimator

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dang, Thi To Nhu; Istas, Jacques. Estimation of the Hurst and the stability indices of a $H$-self-similar stable process. Electron. J. Statist. 11 (2017), no. 2, 4103--4150. doi:10.1214/17-EJS1357. https://projecteuclid.org/euclid.ejs/1508810900


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