Strong laws for quantiles corresponding to moving blocks of random variables (Q1098153)

Let \(U_ 1,U_ 2,..\). be a sequence of independent random variables uniformly distributed on [0,1]. For fixed \(p\in [0,1]\), define \(\zeta_ p(n,k)\) to be the p th quantile on the block \(\{U_{n-k+1},...,U_ n\}\) (1\(\leq k\leq n)\). This paper describes the almost sure limiting behaviour of \(\zeta_ p(n,a_ n)\), where \(a_ n\) is an integer sequence such that \[ \lim_{n\to \infty}a_ n/\log n=\beta \in [0,\infty]. \] For example, if \(\beta =\infty\) and \(n^{-1}a_ n\to a\in [0,1]\), then almost surely \[ \lim_{n\to \infty}\{\max_{a_ n\leq k\leq n}\{\phi (n,k)(\zeta_ p(n,k)-p)\}\}=-(1-(\log a)/4)^{-}. \] \[ Here\quad \phi (n,k)=k^{1/2}(2p(1-p)(\log (n/k)+\log \log k))^{-}. \] The proofs are of two types. The first exploits an approximation of the moving quantile by a lag sum, the strong approximation of \textit{J. Komlós}, \textit{P. Major} and \textit{G. Tusnády} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 33-58 (1976; Zbl 0307.60045)], and results on the increments of Wiener processes by \textit{D. L. Hanson} and the author [e.g. Ann. Probab. 11, 1009-1015 (1983; Zbl 0521.60033)]. The second type of proof is a careful analysis of the probabilities of certain key events.