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Almost sure limit sets of random samples in \(\mathbb{R}^d\). (English) Zbl 0656.60026

Let \(X_1,X_2,\ldots\) be i.i.d. random vectors with values in the cone \([x\ge 0] = [0,\infty)^d\) of \(\mathbb{R}^d\) and with identically distributed components. It is shown that if the function \(R(x)=-\log P[X_1\ge x]\) is regularly varying on \([0,\infty)^d\setminus \{0\}\), in the multivariate sense, and if the limit function \(\lambda (x)=\lim_{t\to \infty}(R(tx)/R(t1))\) is strictly increasing and satisfies \(\lambda (tx)=t^{\alpha}\lambda (x)\) with positive \(\alpha\), then there exist scaling constants \(b_n\) such that the set \(\{b_n^{-1} X_i: 1\le i\le n\}\) converges almost surely to the set \(\{x\ge 0: \lambda(x)\le 1\}\), in the space of compact subsets of \(\mathbb{R}^d\).
This generalises an old result of L. Fisher [see Ann. Math. Stat. 40, 1824–1832 (1969; Zbl 0183.47501)]. Some variants of this theorem, and a partial converse, are also established (including a result in terms of probability densities) and the degree to which various assumptions are essential is explored by means of numerous examples.

MSC:

60D05 Geometric probability and stochastic geometry
60F15 Strong limit theorems

Citations:

Zbl 0183.47501
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