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

Martin J. B. Appel, Michael J. Klass and Ralph P. Russo
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Martin J. B. Appel: United Technologies Research Center
Michael J. Klass: University of California
Ralph P. Russo: University of Iowa

Journal of Theoretical Probability, 1999, vol. 12, issue 1, 27-47

Abstract: 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 ≤i≠j≤n}. 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 (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1023/A:1021788309026

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