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Expectation and variance of the volume covered by a large number of independent random sets. (English) Zbl 0546.60015
Let $$A\subset {\mathbb{R}}^ m$$ be a bounded and measurable set and let $$u_ 1,u_ 2,...\in {\mathbb{R}}^ m$$ be a sequence of i.i.d. random vectors with common probability density $$p(u)=f(\| u\|)$$. Let N be either a nonrandom positive integer or a Poisson random variable with parameter $$\lambda$$. The volume covered by at least (exactly) k of the random sets $$A+u_ i$$, $$i=1,...,N$$, is denoted by $$V_ k (W_ k)$$. In this paper, the asymptotic behaviour of the expectations $$EV_ k$$, $$EW_ k$$ and variances $$\sigma^ 2(V_ 1)$$, as $$\lambda \to\infty$$ or $$N\to\infty$$, is studied for three types of densities p.
The results are far-reaching generalizations of formulas of P. A. P. Moran, Acta Math. 133(1974), 273-286 (1975; Zbl 0297.60011).
Reviewer: W.Weil

##### MSC:
 60D05 Geometric probability and stochastic geometry 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
##### Keywords:
random sets; random covering; asymptotic expansion
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##### References:
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