Journal of Applied Probability

Poisson superposition processes

Harry Crane and Peter McCullagh

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Superposition is a mapping on point configurations that sends the n-tuple (x1, . . ., xn) ∈ Xn into the n-point configuration {x1, . . ., xn} ⊂ X, counted with multiplicity. It is an additive set operation such that the superposition of a k-point configuration in Xn is a kn-point configuration in X. A Poisson superposition process is the superposition in X of a Poisson process in the space of finite-length X-valued sequences. From properties of Poisson processes as well as some algebraic properties of formal power series, we obtain an explicit expression for the Janossy measure of Poisson superposition processes, and we study their law under domain restriction. Examples of well-known Poisson superposition processes include compound Poisson, negative binomial, and permanental (boson) processes.

Article information

J. Appl. Probab., Volume 52, Number 4 (2015), 1013-1027.

First available in Project Euclid: 22 December 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes

Poisson point process permanental process compound Poisson process negative binomial distribution Poisson superposition


Crane, Harry; McCullagh, Peter. Poisson superposition processes. J. Appl. Probab. 52 (2015), no. 4, 1013--1027. doi:10.1239/jap/1450802750.

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