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Strong limit theorems for weighted quantile processes. (English) Zbl 0659.60052
This paper provides a thorough description of the almost sure behaviour of weighted uniform quantile processes. The results are complete analogues of known results on weighted uniform empirical processes. Let \(U_ 1,U_ 2,..\). be a sequence of independent uniform (0,1) random variables. For each integer \(n\geq 1\), let \[ U_ n(s)=U_{k,n},\quad (k-1)/n<s\leq k/n; \] \[ \beta_ n(s)=n^{1/2}\{U_ n(s)-s\},\quad 0\leq s\leq 1;\quad and\quad {\tilde \beta}_ n(s)=\beta_ n(s)\quad for\quad 1\leq (n+1)s\leq n,\quad and\quad =0\quad elsewhere. \] Here \(U_{k,n}\) is the \(k^{th}\) largest value among \(U_ 1,...,U_ n\). Let \[ q\in Q^*=\{q:\quad [0,1]\Rightarrow [0,\infty],\quad nondecrea\sin g,\quad positive\}. \] Theorem 1 states that under suitable conditions on q, there is a functional law of the iterated logarithm for \({\tilde \beta}{}_ n\). For example, if \[ \lim_{s\downarrow 0}(s \log \log (1/s))^{1/2}/q(s)=p=0, \] then with probability one, \(\{{\tilde \beta}_ n/(2\log \log n)^{1/2}q:\quad n=1,2,...\}\) is relatively compact in B[0,1], the space of bounded real-valued functions on [0,1], and has limit set \[ \{f\in B[0,1]:\quad f(0)=f(1)=0,\quad f\quad is\quad absolutely\quad continuous\quad and\quad \int^{1}_{0}(f'(s))^ 2ds\leq 1\}. \] If \(p>0\), then relative compactness of the sequence fails.
Theorem 2 gives the behaviour of \(\beta_ n\) for a certain subclass of weight functions for which \(p=+\infty.\)
Theorem 3 provides LIL-type results for the truncated tail quantile process (for \(k_ n\uparrow \infty\) suitably chosen) \[ \tilde V_ n(s)=(n/k_ n)^{1/2}{\tilde \beta}_ n(sk_ n/n),\quad 0\leq s\leq 1. \] Strong approximations for the truncated quantile process, \({\hat \beta}\), by sums of Brownian bridges, and of the truncated tail quantile process, \(\tilde V,\) by sums of Wiener processes are given in Theorem 4.
Theorem 5 extends J. Kiefer’s [see Nonparametric techniques in statistical inference. Proc. Symp., Indiana Univ., Bloomington 1969, 299- 319 (1970)] results on the supremum of the Bahadur process. The proofs of these results build upon existing results in the field, and require a delicate treatment of asymptotic bounds.
Reviewer: A.Dabrowski

MSC:
60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
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