The law of large numbers for the measure of a region covered by a flow of random sets. (English. Ukrainian original) Zbl 0926.60011

Yurachkivskij, A. P.

Theory Probab. Math. Stat. 55, 187-192 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 173-177 (1996).

Let \(\chi_{ni}(x)\) be a sequence of \(\{0,1\}\)-valued functions determined on the space \((X_n,{\mathcal U}_n,\mu_n)\), \(n\geq 1\), with finite measure and adapted to filtration \(({\mathcal F}_{ni})\). Let \[ \xi_n^{(r)}(t)=\mu_n(t)\{\sum_{i\leq nt}\chi_{ni}\geq r\}/\mu_{n}(X_n). \] The author investigates the asymptotic behaviour of the process \(\xi_n^{(r)}(t)\) as \(n\to\infty\) and proves under some conditions that \(\xi_n^{(r)}(t)\) in \(C(R_{+})\) weakly converges to \(1-\varepsilon^{-Q(t)}\sum_{i=1}^{r-1} Q^i(t)/i!\), where \[ Q(t)=\lim(\mu_n(X_n))^{-1}\sum_{i\leq nt}\int E[\chi_{ni+1}\mid {\mathcal F}_{ni}]d\mu_n. \] {}.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60A10	Probabilistic measure theory
60F05	Central limit and other weak theorems

Keywords:

space with a measure; weak convergence; ergodic theorem