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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)
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