Stationary random point processes with respect to the action of transformation groups in bounded spaces. (English. Ukrainian original) Zbl 0939.60032

Petunin, Yu. I.; Semejko, M. G.

Theory Probab. Math. Stat. 58, 139-149 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 128-138 (1998).

The authors continue their investigation of the geometric theory of random point processes. The main goal of this paper is the definition and investigation of the concept of stationarity of simple point processes. The concept of a bounded space with the action of a group of transformations is introduced. Some types of stationarity are studied. For example one presents the following definition. A simple point process \({\tilde A}=({\mathcal E},X,P)\) in bounded space \((X,U_{X}, {\mathcal B}_{X},G)\) with the action of a group \(G\) is called strongly stationary with respect to the group \(G\) if the measure \(P\) is invariant with respect to the induced group \({\tilde G}\colon P \{{\tilde g}(W)\}=P\{ W\}\) (\(P\{ g(W)\}=P\{ W\}\)) \(\forall{\bar g}\in {\tilde G}\), \(g\in G\), \(W\in X.\) The authors investigate various properties of these processes.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60G10	Stationary stochastic processes

Keywords:

random point process; transformation group; measurable space; stochastic geometry; metric space