A series criterion for the almost-sure growth rate of the generalized diameter of an increasing sequence of random points

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 x S with global maximum (h) over bar. Let {x(n)} be any nondecreasing real sequence converging to (h) over bar. Then p=P(H-n>x(n), infinitely often) is either zero or one, where H-n = max {h(U-i, U-j), 1 less than or equal to i not equal j less than or equal to n}. This paper provides a nonrandom series criterion which is nec essary and sufficient to determine the value of p. In addition, various suf ficient conditions are presented which may be easier to apply. A number of metric space applications are given.