Bridging the gap between a stationary point process and its Palm distribution

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Abstract

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 Po as a limit of conditional probability measures P° N The total variation distance between P° and P1 can be expressed in terms of the P-distribution function of the forward recurrence time.
Original languageEnglish
Pages (from-to)37 - 62
Number of pages26
JournalStatistica Neerlandica
Volume48
Issue number1
Publication statusPublished - 1994

Keywords

  • Radon-Nikodym derivative, local characterization, inversion formula, ergodicity

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