= Exponential void probabilities characterize a Poisson random measure
A <simple locally finite point measure> on $[0,\infty)$ with void <probability> $e^{-\lambda(A)}$ on finite unions of bounded intervals is a <Poisson random measure> with <Lebesgue measure> intensity. Void-event intersections factor, hence the occupation indicators on disjoint cells are independent. The <dyadic occupancy approximation of a simple point measure> is binomial with rare-event <probability> $1-e^{-2^{-n}}$ and converges to a Poisson count of mean the interval length. <Independence> survives joint convergence for disjoint intervals. Extension of the mean <measure> and the <Laplace functional of a Poisson random measure> to Borel functions identifies all Borel counts, not just intervals.
Back to article page
