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

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] Abry, P., Delbeke, L. and Flandrin, P. (1999). Wavelet-based estimators for the self-similarity parameter of $\alpha$-stable processes. In, IEEE International Conference on Acoustics, Speech and Signal Processing III. Az, USA: Phoenix, 1581–1584.
  • [2] Abry, P., Pesquet-Popescu, B. and Taqqu, M. S. (1999). Estimation ondelette des paramètres de stabilité et d’autosimilarité des processus $\alpha-$ stables autosimilaires. In, 17ème Colloque sur le traitement du signal et des images, FRA, 1999. GRETSI, Group d’Etudes du Traitement du Signal.
  • [3] Ayache, A. and Hamonier, J. (2013). Linear fractional stable motion: a wavelet estimator of the alpha parameter., Statistics and Probability Letters 82(8) 1569–1575.
  • [4] Ayache, A. and Hamonier, J. (2015). Linear multifractional stable motion: wavelet estimation of H(.) and $\alpha$ parameter., Lithuanian Mathematical Journal 55(2) 159–192.
  • [5] Bardet, J. M. and Tudor, C. (2010). A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter., Stochastic Processes and their Applications 120(12) 2331–2362.
  • [6] Benassi, A., Bertrand, P., Cohen, S. and Istas, J. (2000). Identification of the Hurst index of a step fractional Brownian motion., Statistical Inference for Stochastic Processes 3(1) 101–111.
  • [7] Benassi, A., Cohen, S. and Istas, J. (1998). Identifying the multifractional function of a Gaussian process., Statistics and Probability Letters 39(4) 337–345.
  • [8] Coeurjolly, J. F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths., Statistical Inference for Stochastic Processes 4(2) 199–227.
  • [9] Cohen, S. and Istas, J. (2013)., Fractional fields and applications. Springer-Verlag, Berlin.
  • [10] Dacunha-Castelle, D. and Duflo, M. (1983)., Probabilités et Statistiques, volume 2. Masson, Paris.
  • [11] Istas, J. (2012). Estimating self-similarity through complex variations., Electronic Journal of Statistics 6 1392–1408.
  • [12] Istas, J. (2012). Manifold indexed fractional fields., ESAIM: Probability and Statistics 16 222–276.
  • [13] Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the Höder index of a Gaussian process., Annales de l’Institut Henri Poincaré (B) Probability and Statistics 33(4) 407–436.
  • [14] Lacaux, C. and Loubes, J.-M. (2007). Hurst exponent estimation of fractional Lévy motions., ALEA - Latin American Journal of Probability and Mathematical Statistics 3 143–161.
  • [15] Le Guével, R. (2013). An estimation of the stability and the localisability functions of multistable processes., Electronic Journal of Statistics 7 1129–1166.
  • [16] Lehmann, E. L. (1999)., Elements of large-sample theory. Springer-Verlag, New York.
  • [17] McCulloch, J. H. (1996)., Financial applications of stable distributions, volume 14. In G. S. Maddala and C. R. Rao, Handbook of Statistics edition, North-Holland, New York.
  • [18] Nourdin, I., Nualart, D. and Tudor, C. (2010). Central and non-central limit theorems for weighted power variations of fractional Brownian motion., Annales de l’Institut Henri Poincaré (B) Probability and Statistics 46(4) 1055–1079.
  • [19] Nourdin, I. and Peccati, G. (2012)., Normal approximations with Malliavin calculus: from Stein’s method to universality, volume 192. Cambridge University Press, Cambridge.
  • [20] Pipiras, V., Taqqu, M. S. and Abry, P. (2007). Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation., Bernoulli 13(4) 1091–1123.
  • [21] Samorodnitsky, G. and Taqqu, M. S. (1994)., Stable Non-Gaussian Random Processes. Chapmann and Hall, New York.
  • [22] Schwartz, L. (1978)., Théorie des distributions. Hermann, Paris.
  • [23] Stoev, S., Pipiras, P. and Taqqu, M. S. (2002). Estimation of the self-similarity parameter in linear fractional stable motion., Signal Processing 82 1873–1901.
  • [24] Stoev, S. and Taqqu, M. S. (2005). Asymptotic self-similarity and wavelet estimation for long-range dependent fractional autoregressive integrated moving average time series with stable innovations., Journal of Time Series Analysis 26(2) 211–249.
  • [25] Takenaka, S. (1991). Integral-geometric construction of self-similar stable processes., Nagoya Mathematical Journal 123 1–12.
  • [26] Van Der Vaart, A.W. (2000)., Asymptotic statistics. Cambrigde University Press, Cambridge.