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Regular stability of large order statistics

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Abstract

For r [greater-or-equal, slanted] 1, let Mnr be the rth largest of {X1, ..., Xn}, where X1,X2,... are i.i.d. random variables with common distribution function F such that F(x) 1 in probability and (*)[mu](n1)/[mu](n) --> h(t) 1. Several sets of necessary and sufficient conditions for regular stability are presented. In particular, {Mnr} is regularly stable if and only if (*) holds, and regular stability with h(t) [reverse not equivalent] 1 is tantamount to complete stability. Moreover, (*) always implies the almost sure stability of {Mnr}, r [greater-or-equal, slanted] 1.

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