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A Series Criterion for the Almost-Sure Growth Rate of the Generalized Diameter of an Increasing Sequence of Random Points

Authors:	Martin J B Appel Michael J Klass Ralph P Russo

Institution:	(1) United Technologies Research Center, East Hartford, Connecticut, 06108;(2) Departments of Statistics and Mathematics, University of California, Berkeley, California, 94720;(3) Department of Statistics and Actuarial Science, University of Iowa, Iowa City, Iowa, 52242

Abstract:	Let U ₁, U ₂,... be a sequence of i.i.d. random mappings taking values in a space S and let h be a symmetric function on S×S with global maximum $\overline h$ Let {x _n} be any nondecreasing real sequence converging to $\overline h$ Then p=P(H _n>x _n, infinitely often) is either zero or one, where H _n=max{h(U _i, U _j), 1 ijn}. This paper provides a nonrandom series criterion which is necessary and sufficient to determine the value of p. In addition, various sufficient conditions are presented which may be easier to apply. A number of metric space applications are given.

Keywords:	Series criteria maximum distance diameter probability on metric spaces
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