Absolute regularity and ergodicity of Poisson count processes

TL;DR

This paper proves that certain Poisson count processes are uniquely stationary, absolutely regular, and ergodic under specific conditions, enabling the adaptation of existing test methods for dependent data.

Contribution

It establishes conditions for stationarity, regularity, and ergodicity of observation-driven Poisson count processes, extending the applicability of testing methods to dependent data.

Findings

Unique stationary distribution established

Process shown to be absolutely regular

Ergodicity demonstrated for the process

Abstract

We consider a class of observation-driven Poisson count processes where the current value of the accompanying intensity process depends on previous values of both processes. We show under a contractive condition that the bivariate process has a unique stationary distribution and that a stationary version of the count process is absolutely regular. Moreover, since the intensities can be written as measurable functionals of the count variables, we conclude that the bivariate process is ergodic. As an important application of these results, we show how a test method previously used in the case of independent Poisson data can be used in the case of Poisson count processes.