## 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.

### MSC:

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

### Citations:

Zbl 0227.62012; Zbl 0664.60038; Zbl 0659.60052; Zbl 0624.60045
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