Applications of the perturbation formula for Poisson processes to elementary and geometric probability
Guenter Last, Sergei Zuyev
TL;DR
The paper develops a unified perturbation framework for point processes, showing that derivatives of event probabilities can be interpreted as counts of pivotal elements via Margulis–Russo-type formulas. Building on this, it presents a perturbation formula for Poisson processes and derives distributional integral identities for classical univariate distributions (Poisson, Erlang, and compound Poisson). It then extends these ideas to multivariate strictly α-stable laws through LePage representations, obtaining new integro-differential equations for their densities. Finally, the work generalizes Crofton’s derivative formula to Poisson and binomial point processes, providing probabilistic proofs and geometric-analytic expressions that connect domain boundary geometry with stochastic perturbations. Across, derivatives acquire a clear probabilistic interpretation as expected numbers of pivotal configurations, offering new tools for analyzing densities and geometric functionals in stochastic geometry.
Abstract
The binomial, the negative binomial, the Poisson, the compound Poisson and the Erlang distribution do all admit integral representations with respect to its (continuous) parameter. We use the Margulis-Russo type formulas for Bernoulli and Poisson processes to derive these representations in a unified way and to provide a probabilistic interpretation for the derivatives. By similar variational methods, we obtain apparently new integro-differential identities which the density of a strictly $α$-stable multivariate density satisfies. Then, we extend Crofton's derivative formula known in integral geometry to the case of a Poisson process. Finally we use this extension to give a new probabilistic proof of a version of this formula for binomial point processes.