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Stochastic properties of the linear multifractional stable motion

Part of: Stochastic processes Distribution theory - Probability Special 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.
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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 processes 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60K99: None of the above, but in this section 60G17: Sample path properties
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 4 , December 2004 , pp. 1085 - 1115
Copyright
Copyright © Applied Probability Trust 2004 

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Footnotes

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

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