Equivalence of functional limit theorems for stationary point processes and their Palm distributions

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Abstract

Let P be the distribution of a stationary point process on the real line and let P0 be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P 0. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P 1, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.
Original language English 593 - 608 16 Probability Theory and Related Fields 81 Published - 1989

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Palm Distribution
Functional Limit Theorem
Stationary point
Point Process
Stationary Process
Equivalence
Real Line
Probability Measure
Infinity
Point process
Limit theorems

Cite this

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title = "Equivalence of functional limit theorems for stationary point processes and their Palm distributions",
abstract = "Let P be the distribution of a stationary point process on the real line and let P0 be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P 0. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P 1, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.",
author = "G. Nieuwenhuis",
year = "1989",
language = "English",
volume = "81",
pages = "593 -- 608",
journal = "Probability Theory and Related Fields",
issn = "0178-8051",
publisher = "Springer New York",

}

In: Probability Theory and Related Fields, Vol. 81, 1989, p. 593 - 608.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Equivalence of functional limit theorems for stationary point processes and their Palm distributions

AU - Nieuwenhuis, G.

PY - 1989

Y1 - 1989

N2 - Let P be the distribution of a stationary point process on the real line and let P0 be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P 0. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P 1, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.

AB - Let P be the distribution of a stationary point process on the real line and let P0 be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P 0. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P 1, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.

M3 - Article

VL - 81

SP - 593

EP - 608

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

ER -