### Abstract

Original language | English |
---|---|

Pages (from-to) | 1612 - 1636 |

Number of pages | 25 |

Journal | Bernoulli |

Volume | 19 |

Issue number | 5A |

DOIs | |

Publication status | Published - Nov 2013 |

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### Keywords

- point process
- Palm distributions
- stationarity
- nonstationarity
- asymptotic mean stationarity
- absolute continuity
- Radon–Nikodym approach
- inversion formulae

### Cite this

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*Bernoulli*, vol. 19, no. 5A, pp. 1612 - 1636. https://doi.org/10.3150/12-BEJ423

**Asymptomatic mean stationarity and absolute continuity of point process distributions.** / Nieuwenhuis, G.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Asymptomatic mean stationarity and absolute continuity of point process distributions

AU - Nieuwenhuis, G.

PY - 2013/11

Y1 - 2013/11

N2 - This paper relates – for point processes Φ on 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 Φ is AMS under the time-shifts if and only if it is AMS under the event-shifts. 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 well-known Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.

AB - This paper relates – for point processes Φ on 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 Φ is AMS under the time-shifts if and only if it is AMS under the event-shifts. 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 well-known Palm relationship for general nonstationary point processes into expressions which resemble results from stationary point process theory.

KW - point process

KW - Palm distributions

KW - stationarity

KW - nonstationarity

KW - asymptotic mean stationarity

KW - absolute continuity

KW - Radon–Nikodym approach

KW - inversion formulae

U2 - 10.3150/12-BEJ423

DO - 10.3150/12-BEJ423

M3 - Article

VL - 19

SP - 1612

EP - 1636

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 5A

ER -