On boundary crossings of the sample \(p\)th quantile (Q1187532)

For \(0<p<1\), define \(\{\xi_{p,n}, n\geq 1\}\) to be the sample \(p\)-th quantiles of an i.i.d. sequence with distribution function \(F\). Assume that a number \(\xi_ p\) exists such that \(F(\xi_ p)=p\) and \(F'(\xi_ p)>0\). Then define \(\theta_ p=(F'(\xi_ p))^ 2/(2p(1-p))\). Suppose \(b(x)\) is a non-increasing differentiable function such that \(xb^ 2(x)/\log\log x\) is non-decreasing with limit \(\omega\), such that \(\omega\phi_ p>1\), as \(x\to\infty\). By a theorem of \textit{L. de Haan} [Ann. of Statist. 2, 815-818 (1974; Zbl 0315.62021)], \[ L(b)=\sup\{n: | \xi_{p,n}-\xi_ p| > b(n)\} \] and \[ N(b)=\sum^ \infty_{n=1} I(|\xi_{p,n}-\xi_ p| > b(n)) \] are random variables. Define \(\theta_ p'=\theta_ p\) if \(\omega=+\infty\); \(\theta_ p'=\theta_ p-\omega^{-1}\) otherwise. Then it is shown that \[ Eg_ r(L(b))< \infty\quad\text{if and only if}\quad \theta\leq r<\theta_ p', \] where \(g_ r(x)=\exp \{rxb^ 2(x)\}\), provided that (i) \(xb^ 2(x)/\log x\downarrow 0\) or (ii) \(b(x)\to 0\) and \(xb^ 2(x)/\log x\) is non-decreasing. Similar results are established for \(N(b)\), and for one-sided versions of \(L(b)\) and \(N(b)\). This work extends some results of the second author and \textit{C.-C. Chao} [Stat. Probab. Lett. 6, 117-123 (1987; Zbl 0634.60027)].