On boundary crossings of the samplepth quantile

Abstract

LetX ₁,X ₂,... be a sequence of independent random variables with distributionF. Suppose that 0<p<1, thatξ _p is the uniquepth quantile ofF, and thatξ _p,n is the samplepth quantile ofX ₁,...,X _n . Ifb(n)→0⁺ sufficiently slowly, then

$$N(b) = \sum\limits_{n = 1}^\infty {I\left\{ {\left| {\xi _{p,n} - \xi _p } \right| > b(n)} \right\}} $$

and

$$L(b) = \sup \left\{ {n:\left| {\xi _{p,n} - \xi _p } \right| > b(n)} \right\}$$

are proper random variables (finite with probability one). In this paper we investigate the moment behavior of exp{Nb ² (N)} and exp{Lb ² (L)}.