Functional central limit theorem for the measure of domain covered by the flow of random sets. (Ukrainian, English) Zbl 1050.60041

Yurachkivs’kyj, A. P.

Teor. Jmovirn. Mat. Stat. 67, 151-160 (2002); translation in Theory Probab. Math. Stat. 67, 169-179 (2003).

Let \(\xi_n(t)\) be the measure of the domain which is not covered by the flow of random subsets of some set up to moment \(t\). Denote by \(\xi^*_n(t)\) the compensator of \(\xi^*_n\). Under certain additional assumptions the author proves that the sequence of random processes \(Y(t)=n^{1/2}(\xi_n-\xi^*_n)\) weakly converges (in \(C\)) to a continuous martingale with deterministic quadratic characteristic (the corresponding explicit expression is given). The proof of the main statement is based on the useful functional CLT for martingales.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60D05	Geometric probability and stochastic geometry
60G42	Martingales with discrete parameter

Keywords:

flow of random sets; measure; functional central limit theorem; martingale; compensator