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.
Similar content being viewed by others
references
Adams, R. A. (1975). Sobolev Spaces, Academic Press, New York.
Burnecki, K., Rosianski, J., and Weron, A. (1998). Spectral representation and structure of stable self-similar processes. In Karatzas, I., Rajput, B. S., and Taqqu, M. S. (eds.), Stochastic Processes and Related Topics: In Memory of Stamatis Cambanis 1943–1995, Trends in Mathematics, Birkhäuser, pp. 1-14.
Chi, Z. (2001). Construction of stationary self-similar generalized fields by random wavelet expansion. Probab. Theory Related Fields 121(2), 269-300.
Hardin, Jr., C. D. (1982). On the spectral representation of symmetric stable processes. J. Multivariate Anal. 12, 385-401.
Mori, T., and Sato, Y. (1996). Construction and asymptotic behavior of a class of self-similar stable processes. Japanese J. Math. 22(2), 275-283.
Pipiras, V., and Taqqu, M. S. (2000). The limit of a renewal-reward process with heavy-tailed rewards is not a linear fractional stable motion. Bernoulli 6(4), 607-614.
Pipiras, V., and Taqqu, M. S. (2002). Decomposition of self-similar stable mixed moving averages. Probab. Theory Related Fields 123(3), 412-452.
Pipiras, V., and Taqqu, M. S. (2002). The structure of self-similar stable mixed moving averages. Ann. Probab. 30(2), 898-932.
RosiAnski, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23(3), 1163-1187.
RosiAnski, J. (1998). Minimal integral representations of stable processes. Preprint.
Runst, T., and Sickel, W. (1996). Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter &; Co., Berlin.
Samorodnitsky, G., and Taqqu, M. S. (1990). (1/α)-self-similar processes with stationary increments. J. Multivariate Anal. 35, 308-313.
Samorodnitsky, G., and Taqqu, M. S. (1994). Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Chapman &; Hall, New York, London.
Surgailis, D., RosiAnski, J., Mandrekar, V., and Cambanis, S. (1998). On the mixing structure of stationary increment and self-similar Sα S processes. Preprint.
Takenaka, S. (1991). Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123, 1-12.
Triebel, H. (1983). Theory of Function Spaces, Birkhäuser Verlag, Basel.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pipiras, V., Taqqu, M.S. Dilated Fractional Stable Motions. Journal of Theoretical Probability 17, 51–84 (2004). https://doi.org/10.1023/B:JOTP.0000020475.95139.37
Issue Date:
DOI: https://doi.org/10.1023/B:JOTP.0000020475.95139.37