Merzbach, ElyNualart, David2015-03-122015-03-121986-02-06Merzbach, 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.https://hdl.handle.net/1808/17066This is the published version, also available here: http://dx.doi.org/10.1214/aop/1176992378.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.point processpoissontwo-parameter processesmartingaleintensitychanging timestopping lineArticle10.1214/aop/1176992378openAccess