Asymptotic mean stationarity and absolute continuity of point process distributions
Abstract
This paper relates  for point processes $\Phi$ on $\mathbb{R}$  two types of asymptotic mean stationarity (AMS) properties and several absolute continuity results for the common probability measures emerging from point process theory. It is proven that $\Phi$ is AMS under the timeshifts if and only if it is AMS under the eventshifts. The consequences for the accompanying two types of ergodic theorem are considered. Furthermore, the AMS properties are equivalent or closely related to several absolute continuity results. Thus, the class of AMS point processes is characterized in several ways. Many results from stationary point process theory are generalized for AMS point processes. To obtain these results, we first use Campbell's equation to rewrite the wellknown Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.2726
 Bibcode:
 2013arXiv1312.2726N
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.3150/12BEJ423 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)