Dilated Fractional Stable Motions

Vladas Pipiras () and Murad S. Taqqu ()
Additional contact information
Vladas Pipiras: University of North Carolina at Chapel Hill
Murad S. Taqqu: Boston University

Journal of Theoretical Probability, 2004, vol. 17, issue 1, 51-84

Abstract: Abstract Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called “random wavelet expansion” processes.

Keywords: Stable; self-similar processes with stationary increments; dilated fractional stable motions; uniqueness; flows; the telecom process (search for similar items in EconPapers)
Date: 2004
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1023/B:JOTP.0000020475.95139.37 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:17:y:2004:i:1:d:10.1023_b:jotp.0000020475.95139.37

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/B:JOTP.0000020475.95139.37

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().