On the Functional Form of the Law of the Iterated Logarithm for the Partial Maxima of Independent Identically Distributed Random Variables

Abstract

Let $k$ be a positive integer, let $X, X_1, X_2, \cdots$ be i.i.d. random variables, and let $m_n^{(k)}$ be the $k$th largest of $X_1, \cdots, X_n$. Let $(M_n^{(k)}(t))_{0 < t < \infty}$ be the random process defined by $M_n^{(k)} (t) = m^{(k)}_{\lbrack nt\rbrack}. M_n^{(k)}$ takes values in the space $D$ of non-decreasing right-continuous functions on $(0, \infty)$. Let $D$ be endowed with the usual topology of weak convergence. We show that if $X$ is uniformly distributed over [-1,0], then wp 1 the sequence $(M_n^{(k)}/(\log_2 n/n))_{n\geqq 3}$ is relatively compact in $D$ and its limit points coincide with $\{x\in D: x(t) \leqq 0$ for all $t$, and $\int x(t) dt \geqq -1\}$. Also, we show that if $X$ is exponential with mean 1, then wp 1 the sequence $((M_n^{(k)} - \log n)/\log_2n)_{n\geqq 3}$ is relatively compact in $D$ and its limit points coincide with $\{x\in D: x(t) \geqq 0$ for all $t$, and $\lambda_k(x) \leqq 1\}$; here $\lambda_k(x) = \sup (\sum_{p < q} x(t_p) + kx(t_q))$, with the supremum being taken over all finite systems of points $\{t_p\}_{p \leqq q}$ over which $x$ is strictly increasing. Extensions of and corollaries to these results are given.