A characterization of the spatial Poisson process and changing time (Q1820505)

\textit{S. Watanabe} [Jap. J. Math. 34(1964), 53-70 (1965; Zbl 0141.157)] proved that a point process \(X_ t\) \((t\in {\mathbb{R}}_+)\) is a Poisson process if and only if the process \(X_ t-t\) is a martingale. This result was generalized by \textit{P. Brémaud} [Point processes and queues. Martingale dynamics. (1981; Zbl 0478.60004)] for doubly stochastic Poisson processes. In this paper these results are extended to two- parameter point processes \(X_ z\) \((z\in {\mathbb{R}}^ 2_+)\). Using this characterization, the problem of transforming a two-parameter point process into a two-parameter Poisson process is studied.