## Ramsey’s theorem and Poisson random measures.(English)Zbl 0533.60057

Let N be a random point process defined on a $$\delta$$-ring $${\mathcal D}$$ of subsets of a measurable space and suppose that N has independent increments: whereas $$D_ 1,...,D_ k\in {\mathcal D}$$ are disjoint the random variables $$N(D_ 1),...,N(D_ k)$$ are independent. Define a set $$D\in {\mathcal D}$$ to be small with respect to N if $$N(D)=0$$ a.s. A. Prékopa [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 1, 153-170 (1958; Zbl 0089.340)] showed that if singletons belong to $${\mathcal D}$$ and are small then necessarily N is a Poisson point process. The authors of this paper obtain the same conclusions under the formally weaker condition that for each $$D\in {\mathcal D}$$ there exist a countable subfamily $${\mathcal B}$$ of $${\mathcal D}$$ such that $$D\subset U\{B:B\in {\mathcal B}\}$$ and for each $$x\in D$$, $$\cap \{B\in {\mathcal B}:x\in B\}$$ is small. Their method of proof is based on appeal to Ramsey’s theorem in combinatorial analysis.
Reviewer: A.Karr

### MSC:

 60G57 Random measures 05C55 Generalized Ramsey theory 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

### Keywords:

Poisson process; Ramsey theorem; independent increments

Zbl 0089.340
Full Text: