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Note on generalizations of Mecke's formula and extensions of H = λG

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Masakiyo Miyazawa
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
*
∗Postal address: Department of Information Science, Science University of Tokyo, Noda, Chiba 278, Japan.
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Abstract

Mecke's formula is concerned with a stationary random measure and shift-invariant measure on a locally compact Abelian group , and relates integrations concerning them to each other. For , we generalize this to a pair of random measures which are jointly stationary. The resulting formula extends the so-called Swiss Army formula, which was recently obtained as a generalization for Little's formula. The generalized Mecke formula, which is called GMF, can be also viewed as a generalization of the stationary version of H = λG. Under the stationary and ergodic assumptions, we apply it to derive many sample path formulas which have been known as extensions of H = λG. This will make clear what kinds of probabilistic conditions are sufficient to get them. We also mention a further generalization of Mecke's formula.

Keywords

PALM CALCULUSSTATIONARY PROCESSRANDOM MEASUREPOINT PROCESSLITTLE'S FORMULA

MSC classification

Primary: 60G57: Random measures
Secondary: 60K25: Queueing theory 60G55: Point processes 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 105 - 122
Copyright
Copyright © Applied Probability Trust 1995 

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Footnotes

Research partly supported by NEC C&C Laboratories.

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