Characterization of time-simple random point measures with stationary independent increments.

Finite-rate mark sets have finite intensity, disjoint boxes have independent Poisson counts, and natural-history compensation holds. The joint complex generating function is proved by integrating a linear remainder estimate over a uniform time grid; intersecting the cells with the boxes handles arbitrary endpoints without a separate time-refinement argument.

The namespace is BGW.StableTrees because this characterization serves the excursion theory of the stable process.

References

• O. Kallenberg, Foundations of Modern Probability, 2nd ed., Springer, 2002, Chapter 12 (Poisson and pure jump-type Markov processes; characterization of Poisson processes).
• K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999, Chapter 1.