Nonstandard Functional Laws of the Iterated Logarithm for Tail Empirical and Quantile Processes

Abstract

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.