Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-13T22:03:07.697Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic properties of the linear multifractional stable motion

Part of:Distribution theory - ProbabilityStochastic processesSpecial processes

Published online by Cambridge University Press:  01 July 2016

Stilian Stoev and
Murad S. Taqqu
Stilian Stoev*
Affiliation:
Boston University
Murad S. Taqqu*
Affiliation:
Boston University
*
∗ Postal address: Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA.
∗ Postal address: Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA.
Get accessRights & Permissions [Opens in a new window]

Abstract

We study a family of locally self-similar stochastic processes Y = {Y(t)}t∈ℝ with α-stable distributions, called linear multifractional stable motions. They have infinite variance and may possess skewed distributions. The linear multifractional stable motion processes include, in particular, the classical linear fractional stable motion processes, which have stationary increments and are self-similar with self-similarity parameter H. The linear multifractional stable motion process Y is obtained by replacing the self-similarity parameter H in the integral representation of the linear fractional stable motion process by a deterministic function H(t). Whereas the linear fractional stable motion is always continuous in probability, this is not in general the case for Y. We obtain necessary and sufficient conditions for the continuity in probability of the process Y. We also examine the effect of the regularity of the function H(t) on the local structure of the process. We show that under certain Hölder regularity conditions on the function H(t), the process Y is locally equivalent to a linear fractional stable motion process, in the sense of finite-dimensional distributions. We study Y by using a related α-stable random field and its partial derivatives.

Keywords

Linear fractional stable motionself-similaritymultifractional Brownian motionlocal self-similaritytangent processheavy tail

MSC classification

Primary:60G18: Self-similar processes60E07: Infinitely divisible distributions; stable distributions
Secondary:60K99: None of the above, but in this section60G17: Sample path properties

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004, pp. 1085 - 1115
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Partially supported by NSF grant DMS-0102410 at Boston University.

References

Ayache, A. and Lévy-Véhel, J. (1999). Generalized multifractional Brownian motion: definition and preliminary results. In Fractals: Theory and Applications in Engineering, eds Dekking, M., Lévy-Véhel, J., Lutton, E. and Tricot, C., Springer, London, pp. 17–32.Google Scholar
Ayache, A. and Lévy-Véhel, J. (2000). The generalized multifractional Brownian motion. Statist. Infer. Stoch. Process. 3, 7–18.Google Scholar
Ayache, A. and Taqqu, M. S. (2003). Multifractional processes with random exponent. Preprint. Available as http://www.cmla.ens-cachan.fr/Cmla/Publications/2003/CMLA2003-19.ps.gz Google Scholar
Bardet, J.-M. and Bertrand, P. (2003). Definition, properties and wavelet analysis of multiscale fractional Brownian motion. Preprint.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (1998). Identifying the multifractional function of a Gaussian process. Statist. Prob. Lett. 39, 337–345.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (2002). Identification and properties of real harmonizable fractional Lévy motions. Bernoulli 8, 97–115.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10, 357–373.CrossRefGoogle Scholar
Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13, 19–90. Statist. Infer. Stoch. Process. 3, 101–111.Google Scholar
Cohen, S. (1999). From self-similarity to local self-similarity: the estimation problem. In Fractals: Theory and Applications in Engineering, eds Dekking, M., Lévy-Véhel, J, Lutton, E and Tricot, C., Springer, London, pp. 3–16.Google Scholar
Cohen, S. and Istas, J. (2004). A universal estimator of local self-similarity. Preprint. Available at http://brassens.upmf-grenoble.fr/∼jistas/publications.html Google Scholar
Falconer, K. J. (2002). Tangent fields and the local structure of random fields. J. Theoret. Prob. 15, 731–750.Google Scholar
Falconer, K. J. (2003). The local structure of random processes. J. London Math. Soc. 67, 657–672.CrossRefGoogle Scholar
Park, K. and Willinger, W. (eds) (2000). Self-Similar Network Traffic and Performance Evaluation. John Wiley, New York.Google Scholar
Paxson, V. and Floyd, S. (1995). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Networking 3, 226–244.Google Scholar
Peltier, R. F. and Lévy-Véhel, J. (1995). Multifractional Brownian motion: definition and preliminary results. Tech. Rep. 2645, INRIA.Google Scholar
Pipiras, V. and Taqqu, M. S. (2004). Stable stationary processes related to cyclic flows. Ann. Prob. 32, 2222–2260.CrossRefGoogle Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York.Google Scholar
Stoev, S. and Taqqu, M. S. (2004). Path properties of the linear multifractional stable motion. To appear in Fractals.Google Scholar
Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1998). On the mixing structure of stationary increment and self-similar SαS processes. Preprint.Google Scholar