S Normal approximation of order statistics. Let .Xi /i1 be a sequence of i.i.d. random variables with common distribution Q on an interval X. Assume that the distribution function F D FQ is differentiable on X with continuous derivative D F 0 > 0. Let 0 < ˛ < 1, q 2 X the associated ˛-quantile of Q, and .jn/ a sequence in N with jjn  ˛nj=

p n ! 0.

Show that p n.XjnWn  q/ !d N0;v as n ! 1, where v D ˛.1˛/= .q/2. Hint: Use (8.17) and the fact that Corollary (5.24)

still holds when the underlying success probability depends on n but converges to a limit.