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October, 1990 Nonstandard Functional Laws of the Iterated Logarithm for Tail Empirical and Quantile Processes
Paul Deheuvels, David M. Mason
Ann. Probab. 18(4): 1693-1722 (October, 1990). DOI: 10.1214/aop/1176990642


Let $\{\alpha_n(t), 0 \leq t \leq 1\}$ and $\{\beta_n(s), 0 \leq s \leq 1\}$ denote the uniform empirical and quantile processes. We show that, for suitable sequences $A(n, \kappa_n)$ and $B(n, l_n)$, the tail empirical process $\{A(n, \kappa_n)\alpha_n(n^{-1}\kappa_nt), 0 \leq t \leq 1\}$ and the tail quantile process $\{B(n, l_n)\beta_n(n^{-1}l_n s), 0 \leq s \leq 1\}$ are almost surely relatively compact in appropriate topological spaces, where $0 \leq \kappa_n \leq n$ and $0 \leq l_n \leq n$ are sequences such that $\kappa_n$ and $l_n$ are $O(\log \log n)$ as $n \rightarrow \infty$. The limit sets of functions are defined through integral conditions and differ from the usual Strassen set obtained when $\kappa_n$ and $l_n$ are $\infty(\log \log n)$ as $n \rightarrow \infty$. Our results enable us to describe the strong limiting behavior of classical statistics based on the top extreme order statistics of a sample or on the empirical distribution function considered in the tails.


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Paul Deheuvels. David M. Mason. "Nonstandard Functional Laws of the Iterated Logarithm for Tail Empirical and Quantile Processes." Ann. Probab. 18 (4) 1693 - 1722, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0719.60030
MathSciNet: MR1071819
Digital Object Identifier: 10.1214/aop/1176990642

Primary: 60F15
Secondary: 60F05 , 60F17 , 62G30

Keywords: empirical and quantile processes , Extreme values , Functional laws of the iterated logarithm , large deviations , order statistics , strong laws

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • October, 1990
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