Absolute regularity and ergodicity of Poisson count processes

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.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ313 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics

“…As an example, an L 2 -test for the intensity function of a Poisson count model is discussed in Section 5.3 below. According to Neumann (2011), the underlying process is ergodic but not mixing in general. As in the majority of papers in the literature, we employ a spectral decomposition of the kernel to obtain an additive structure such that we can use a central limit theorem to proceed to the limit.…”

Robust CUSUM test for time series of counts and its application to analyzing the polio incidence data

“…Under the above condition, there is a strictly stationary and ergodic solution for model (2.1) and any order moments of X t and λ t are finite (Neumann, 2011;Doukhan et al, 2012).…”

Integer-Valued APARCH Processes

“…Neumann [12] proved that the contractive condition in (4) is, indeed, sufficient to ensure uniqueness of the stationary distribution and ergodicity of (Y t , λ t ). The results are quoted below.…”

“…Following the work of Doukhan et al [5] (see also [2], [12], [9]) we will establish the existence and uniqueness of a stationary solution, and ergodicity for the p = q = 1 case. The INAPARCH(1, 1) process is defined as an integer-valued process (Y t ) such that…”

“…In proving uniqueness we proceed as follows: first note that the INAPARCH(1, 1) model belongs to the class of observation-driven Poisson count processes considered in Neumann [12]…”