Hostname: page-component-84b7d79bbc-c654p Total loading time: 0 Render date: 2024-07-31T01:31:39.675Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of self-similar random measure

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

David Vere-Jones
David Vere-Jones*
Affiliation:
Victoria University and Statistical Research Associates, Ltd
*
∗ Postal address: Department of Mathematics, Victoria University of Wellington, Wellington, New Zealand. Email address: dvj@mcs.vuw.ac.nz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We describe a class of self-similar random measure that generalizes the class of stable, completely random measure, or, in one dimension, the class of processes with stable independent increments. As with the stable processes, the realizations are purely atomic, but the masses of the atoms are not necessarily independent, but rather characterized by self-similar dependence relations. Indeed, the class can be described most effectively in terms of the point process on the product space for the locations and sizes (‘marksȉ) of the atoms. Then self-similarity reduces to an invariance relation (‘biscale invariance’) on the distribution of this marked point process. The condition can be satisfied when the marked point process is compound Poisson, corresponding to the nonnegative stable processes, but is by no means restricted to this case. An example is given which modifies and extends Ogata's epidemic-type aftershock sequence model for earthquake occurrence.

Keywords

Self-similar processrandom measurepoint processLévy process

MSC classification

Primary: 60G18: Self-similar processes 60G55: Point processes 60G57: Random measures
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 908 - 914
Copyright
Copyright © Applied Probability Trust 2005 

References

Brémaud, P. and Massoulié, L. (2001). Hawkes branching point processes without ancestors. J. Appl. Prob. 38, 122–135.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol I. Elementary Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
Kingman, J. F. C. (1967). Completely random measures. Pacific J. Math. 21, 59–78.Google Scholar
Lamperti, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78.CrossRefGoogle Scholar
Ogata, Y. (1988). Statistical models for earthquake occurrence and residual analysis for point processes. J. Amer. Statist. Assoc. 83, 9–27.Google Scholar
Ogata, Y. (1998). Space-time point process models for earthquake occurrences. Ann. Inst. Statist. Math. 50, 379–402.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
Vervaat, W. (1985). Sample path properties of self-similar processes with stationary increments. Ann. Prob. 13, 1–27.Google Scholar
Zähle, U. (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. Prob. Theory Relat. Fields 80, 79–100.CrossRefGoogle Scholar
Zähle, U. (1990a). Self-similar random measures. II. A generalization to self-affine measures. Math. Nachr. 146, 85–98.Google Scholar
Zähle, U. (1990b). Self-similar random measures. IV. The recursive construction model of Falconer, Graf, and Mauldin and Williams. Math. Nachr. 149, 285–302.Google Scholar
Zähle, U. (1991). Self-similar random measures. III. Self-similar random processes. Math. Nachr. 151, 121–148.Google Scholar