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Bridging the gap between a stationary point process and its Palm distribution
Author(s) -
Nieuwenhuis G.
Publication year - 1994
Publication title -
statistica neerlandica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.52
H-Index - 39
eISSN - 1467-9574
pISSN - 0039-0402
DOI - 10.1111/j.1467-9574.1994.tb01430.x
Subject(s) - mathematics , ergodic theory , probability measure , corollary , stationary ergodic process , point process , conditional probability , probability distribution , conditional probability distribution , combinatorics , discrete mathematics , statistical physics , mathematical analysis , statistics , invariant measure , physics
In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at random. That is, either the stationary distribution P or its Palm distribution P° is the ruling probability measure. In this paper an approach is presented to bridge the gap between these distributions. We consider probability measures which give exactly the same events zero probability as P°, having simple relations with P . Relations between P and P° are derived with these intermediate measures as bridges. With the resulting Radon‐Nikodym densities several well‐known results can be proved easily. New results are derived. As a corollary of cross ergodic theorems a conditional version of the well‐known inversion formula is proved. Several approximations of P° are considered, for instance the local characterization of P o as a limit of conditional probability measures P° N The total variation distance between P° and P 1 can be expressed in terms of the P‐distribution function of the forward recurrence time.