Concerning a new approach to statistical inferences about non-homogeneous Poisson processes with periodic intensity

Abstract

We consider probabilistic properties of the jump times of a non-homogeneous Poisson process. Under the assumption that the instantaneous intensity is a periodic function separated from zero, the “phases” (fractional parts with respect to the period length) of the jump times form a general Markov chain on an interval, and the density of the invariant probability distribution is proportional to the instantaneous intensity function.

Possible applications of the discovered property of the jump times to statistical inference about a non-homogeneous Poisson process are discussed. Using the Central Limit Theorem for Markov chains, we prove the convergence of the finite-dimensional distributions of the empirical process to the finite-dimensional distributions of a Gaussian process of the same type as in the case of an independent repeated sample.