Generalization of one problem of stochastic geometry and related measure-valued processes (Q2722121)

In stochastic geometry one of the interests is the cover problem: find a distribution or mean value of the measure of a field covered by a union of random sets. The following generalization is considered: find a distribution of the measure of a field in which the value of the random field does not exceed a given level (in the cover problem this field is a sum of indicator sets). An explicit expression is not found, but only an asymptotical one. For the application of the proposed approach it is essential that the field depends, besides of space variable, also on continuous time variable. NEWLINENEWLINENEWLINEThe author [Ukr. Math. J. 51, No. 4, 604-617 (1999); translation from Ukr. Mat. Zh. 51, No. 4, 542-552 (1999; Zbl 0957.60027)] proved the functional limit theorem like LLN for general covering processes. The aim of this paper is to illustrate this result with the help of a geometric model. Namely, the functional limit theorem is proved for the measure of a region in which values of time-dependent random fields do not exceed a given level.