Poisson superposition processes

Abstract

Superposition is a mapping on point configurations that sends the n-tuple (x₁, . . ., x_n) ∈ Xⁿ into the n-point configuration {x₁, . . ., x_n} ⊂ 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.