## Electronic Journal of Statistics

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

#### 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
First available in Project Euclid: 24 October 2017

https://projecteuclid.org/euclid.ejs/1508810900

Digital Object Identifier
doi:10.1214/17-EJS1357

Mathematical Reviews number (MathSciNet)
MR3715823

Zentralblatt MATH identifier
06805088

#### 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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