Nonstandard functional laws of the iterated logarithm for tail empirical and quantile processes. (English) Zbl 0719.60030

Let \(\{U_ i:\) \(i=1,2,...\}\) be a sequence of independent random variables uniformly distributed on [0,1]. Set \(U_ n(t)=n^{-1} card\{i:\;U_ i\leq t,\quad 1\leq i\leq n\}\) for \(0\leq t\leq 1\) and \(V_ n(t)=\inf \{t\geq 0:\;U_ n(t)\geq s\}\) for \(0<s\leq 1\) with \(V_ n(0)=0\) (i.e. the empirical distribution and quantile functions). The uniform empirical and quantile processes are \(\alpha_ n(t)=n^{1/2}(U_ n(t)- t)\) and \(\beta_ n(s)=n^{1/2}(V_ n(s)-s)\) for \(0\leq s,t\leq 1\). Functional forms of the law of the iterated logarithm are well-known for both \(\alpha_ n\) and \(\beta_ n\) [see H. Finkelstein, Ann. Math. Statist. 42, 607-615 (1971; Zbl 0227.62012)]. In both cases the limit set consists of all absolutely continuous functions on [0,1] satisfying \(f(0)=f(1)=0\) and \(\int^{\infty}_{0}(\dot f(u))^ 2du\leq 1\). Consider now the strong limiting behaviour of the tail empirical and quantile processes, i.e. let \(0<\kappa_ n\leq n\) be a sequence of real numbers and consider \(\{A(n,\kappa_ n)\alpha_ n(n^{-1}\kappa_ nt):\;0\leq t\leq 1\}\) and \(\{B(n,\kappa_ n)\beta_ n(n^{-1}\kappa_ ns):\;0\leq s\leq 1\},\) where \(A(n,\kappa_ n)\) and \(B(n,\kappa_ n)\) are suitable norming constants. If \(n^{-1}\kappa_ n\to 0\) and \(\kappa_ n/\log \log n\to \infty\) as \(n\to \infty\), then for \(A(n,\kappa_ n)=B(n,\kappa_ n)=(2n^{-1}\kappa_ n \log \log n)^{-1/2}\) the tail empirical and quantile processes are almost surely relatively compact and the set of limit points is the usual set in Strassen’s functional LIL [see D. M. Mason, Ann. Inst. Henri Poincaré, Probab. Stat. 24, No.4, 491-506 (1988; Zbl 0664.60038)] and J. H. J. Einmal and D. M. Mason, Ann. Probab. 16, No.4, 1623-1643 (1988; Zbl 0659.60052)].
The paper under review resolves the question of limit sets when \(n^{- 1}\kappa_ n\to 0\) and either \(\kappa_ n/\log \log n\to c\in (0,\infty)\) or \(\kappa_ n/\log \log n\to 0\) as \(n\to \infty\). In these cases, the limit sets are not those given by Strassen’s or Finkelstein’s results, and are different for \(\alpha_ n\) and \(\beta_ n\). Here we state only one representative theorem (Theorem 2.1) from this paper; certain technical conditions and definitions have been suppressed for the sake of brevity. If \(0\leq \kappa_ n\leq n\) satisfies \(\kappa_ n/\log \log n\to c\in (0,\infty)\) as \(n\to \infty\), then the sequence of functions \(\{\) (log log n)\({}^{-1}(nU_ n(n^{-1}\kappa_ nt)\}\) is almost surely relatively compact with the set of limit points given by all absolutely continuous f on (0,1) with \(f(0)=0\) and \[ \int^{1}_{0}(\dot f(u)\cdot \log (\dot f(u)/c)-\dot f(u)+c)du\leq 1. \] The proofs rely on new results on large deviations due to J. Lynch and J. Sethuraman [Ann. Probab. 15, 610-627 (1987; Zbl 0624.60045)]. The authors apply their results to several classical statistics based on sample extremes.


60F15 Strong limit theorems
60F05 Central limit and other weak theorems
62G30 Order statistics; empirical distribution functions
60F17 Functional limit theorems; invariance principles
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