Dominated Families of Shifted Palm Distributions

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Abstract

In stationary point process theory, the concept Palm distribution plays an important role.Many important results (like for instance Little s law, so important in many fields) arise from it.However, in the non-stationary case a whole family of local Palm distributions (PD s) has to be considered and the concept seems to loose its importance.The present paper mainly considers non-stationary point processes, and studies relations between the distribution P of a point process, the family {Px} of PD s, and the family {P0,x} of shifted PD s.Here P0,x is the probability distribution that is experienced from an occurrence (arrival, point, transaction) at x.It is attempted to regain some of the glance of the concept Palm distribution by considering generalizations of results that are basic for stationary point processes.Starting point is a refined version of Campbell s equation, which expresses the general relationship between the distribution P of the point process and the family {Px} of PD s.It is used to generalize the inversion formula, well known from stationary point process theory.This generalization is basic; it leads to several relations regarding the above distributions.In the second part of the research domination assumptions are imposed: either the null-sets of a time-stationary distribution are also null-sets of P or the nullsets of one event-stationary distribution are also null-sets of almost all shifted PD s.Under such domination regulations, P0,x can explicitly be expressed in terms of P and several stationary-case long-run properties can be generalized.The relationship between the two types of domination assumptions is carefully studied.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages36
Volume2005-7
Publication statusPublished - 2005

Publication series

NameCentER Discussion Paper
Volume2005-7

Fingerprint

Palm Distribution
Point Process
Null set or empty set
Stationary point
Stationary Process
Domination
Stationary Distribution
Nonstationary Processes
Inversion Formula
Long-run
Transactions
Family
Probability Distribution
Express
Generalise

Keywords

  • point processes
  • non-stationarity
  • family of Palm distributions
  • domination

Cite this

Nieuwenhuis, G. (2005). Dominated Families of Shifted Palm Distributions. (CentER Discussion Paper; Vol. 2005-7). Tilburg: Econometrics.
Nieuwenhuis, G. / Dominated Families of Shifted Palm Distributions. Tilburg : Econometrics, 2005. (CentER Discussion Paper).
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Nieuwenhuis, G 2005 'Dominated Families of Shifted Palm Distributions' CentER Discussion Paper, vol. 2005-7, Econometrics, Tilburg.

Dominated Families of Shifted Palm Distributions. / Nieuwenhuis, G.

Tilburg : Econometrics, 2005. (CentER Discussion Paper; Vol. 2005-7).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Dominated Families of Shifted Palm Distributions

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AB - In stationary point process theory, the concept Palm distribution plays an important role.Many important results (like for instance Little s law, so important in many fields) arise from it.However, in the non-stationary case a whole family of local Palm distributions (PD s) has to be considered and the concept seems to loose its importance.The present paper mainly considers non-stationary point processes, and studies relations between the distribution P of a point process, the family {Px} of PD s, and the family {P0,x} of shifted PD s.Here P0,x is the probability distribution that is experienced from an occurrence (arrival, point, transaction) at x.It is attempted to regain some of the glance of the concept Palm distribution by considering generalizations of results that are basic for stationary point processes.Starting point is a refined version of Campbell s equation, which expresses the general relationship between the distribution P of the point process and the family {Px} of PD s.It is used to generalize the inversion formula, well known from stationary point process theory.This generalization is basic; it leads to several relations regarding the above distributions.In the second part of the research domination assumptions are imposed: either the null-sets of a time-stationary distribution are also null-sets of P or the nullsets of one event-stationary distribution are also null-sets of almost all shifted PD s.Under such domination regulations, P0,x can explicitly be expressed in terms of P and several stationary-case long-run properties can be generalized.The relationship between the two types of domination assumptions is carefully studied.

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BT - Dominated Families of Shifted Palm Distributions

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Nieuwenhuis G. Dominated Families of Shifted Palm Distributions. Tilburg: Econometrics. 2005. (CentER Discussion Paper).