zbMATH — the first resource for mathematics

Covaristic functions of random measures and their applications. (English. Ukrainian original) Zbl 0955.60051
Theory Probab. Math. Stat. 60, 187-200 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 167-176 (1999).
The article deals with the covaristic (covariation characteristic) functions which were introduced by the author [Ukr. Math. J. 51, No. 4, 604-617 (1999); translation from Ukr. Mat. Zh. 51, No. 4, 542-552 (1999)]. Let \(G=G(R^{d})\) be the space of all probabilistic measures on \({\mathcal B}^{d}.\) Then a random measure is a \(G\)-valued random element. The characteristic function of a random measure \(\psi(\nu)\) is a random function. The author calls the function \[ {\widetilde\psi}(l,z_1,z_2,\ldots)=E\prod\limits_{j=1}^l \int e^{iz_{j}v} \psi(d\psi) \] covaristic function of a random measure. The author proves that the distribution of a random measure is determined by its covaristic function. Furthermore, he proves other properties of covaristic functions, for example, the Lévy theorem for covaristic functions. An example of estimation of covaristic functions is presented. Some applications of covaristic functions to stochastic geometry are considered.

60G57 Random measures
60D05 Geometric probability and stochastic geometry