Characterization of nonhomogeneous Poisson processes via moment conditions

Abstract

The property of independent increments is one of the most important for defining both the homogeneous and nonhomogeneous Poisson process. In this paper we give two ways to relax this requirement and characterize the nonhomogeneous Poisson process by some moment conditions. One result is that a counting process { N( t), t ⩾ 0} with finite moments of all orders is a nonhomogeneous Poisson process with mean functions m( t) = EN( t) if and only if for any t i , i = 1,…,, k, cum( N( t 1),…, N( t k ) = min 1 ⩽ i ⩽ k EN( t i ), where cum(·) is the joint multivariate cumulant. A second result is that if increments on any inteval are Poisson distributed and an exchangeable condition is assumed then the process is nonhomogeneous Poisson. This extends Renyi's (1967) result.