Mjb. Appel et al., A series criterion for the almost-sure growth rate of the generalized diameter of an increasing sequence of random points, J THEOR PR, 12(1), 1999, pp. 27-47
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.