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October, 1986 A Characterization of the Spatial Poisson Process and Changing Time
Ely Merzbach, David Nualart
Ann. Probab. 14(4): 1380-1390 (October, 1986). DOI: 10.1214/aop/1176992378

Abstract

Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.

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Ely Merzbach. David Nualart. "A Characterization of the Spatial Poisson Process and Changing Time." Ann. Probab. 14 (4) 1380 - 1390, October, 1986. https://doi.org/10.1214/aop/1176992378

Information

Published: October, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0615.60047
MathSciNet: MR866358
Digital Object Identifier: 10.1214/aop/1176992378

Subjects:
Primary: 60G55
Secondary: 60G40 , 60G48 , 60G60

Keywords: changing time , intensity , martingale , point process , Poisson , stopping line , two-parameter process

Rights: Copyright © 1986 Institute of Mathematical Statistics

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Vol.14 • No. 4 • October, 1986
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