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A limit theorem with application to Båth's law in seismology

Part of:Special processesLimit theoremsGeophysics Geophysics (general)

Published online by Cambridge University Press:  01 July 2016

David Vere-Jones
David Vere-Jones*
Affiliation:
Victoria University of Wellington and Statistical Research Associates Ltd
*
∗ Postal address: School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, PO Box 600, Wellington, 6140, New Zealand. Email address: david.vere-jones@mcs.vuw.ac.nz
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Abstract

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In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

Keywords

Extreme value distributionETAS modelBåth's lawPoisson limitsexponential variates

MSC classification

Primary:60F05: Central limit and other weak theorems60K40: Other physical applications of random processes
Secondary:86A15: Seismology

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008, pp. 882 - 896
Copyright
Copyright © Applied Probability Trust 2008 

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