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A Characterization of the Spatial Poisson Process and Changing Time
Merzbach, Ely Nualart, David
Merzbach, Ely
Nualart, David
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Abstract
Watanabe proved that if Xt is a point process such that Xt−t is a martingale, then Xt 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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This is the published version, also available here: http://dx.doi.org/10.1214/aop/1176992378.
Date
1986-02-06
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Institute of Mathematical Statistics (IMS)
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NualartD_14(4)1380.pdf
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Keywords
point process, poisson, two-parameter processes, martingale, intensity, changing time, stopping line
Citation
Merzbach, Ely; Nualart, David. A Characterization of the Spatial Poisson Process and Changing Time. Ann. Probab. 14 (1986), no. 4, 1380--1390. http://dx.doi.org/10.1214/aop/1176992378.