Stability theorems for large order statistics with varying ranks

Let {Xn}; be a sequence of i.i.d. random variables with distribution function F. For r [greater-or-equal, slanted] 1 and n [greater-or-equal, slanted] r, let Zrn be the rth largest of };X1,...,Xn};. Let [mu]n = F-1(1 - n-1), n [greater-or-equal, slanted] 1, and suppose that {rn}; is a non-decreasing integer sequence such that 1 [less-than-or-equals, slant] rn} [less-than-or-equals, slant] n. It is shown that Zrnn/[mu]n --> 1 in probability if and only if (*) rn/(n(1 - F([delta][mu]n))) --> 0 for every [delta] < 1. Moreover, if Zrn/[mu]n --> 1 a.s. for some r [greater-or-equal, slanted] 1, then Zrnn/[nu]n --> 1 a.s. if and only if (*) holds. These results generalize a theorem of Hall (1979).