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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: [email protected]
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Abstract

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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 

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