Linear Bounds on the Empirical Distribution Function

Abstract

Let $\Gamma_n$ denote the empirical df of a sample from the uniform (0, 1) df $I$. Let $\xi_{nk}$ denote the $k$th smallest observation. Let $\lambda_n > 1$. Let $A_n$ denote the event that $\Gamma_n$ intersects the line $\lambda_n I$ on [0, 1] and let $B_n$ denote the event that $\Gamma_n$ intersects the line $I/\lambda_n$ on $\lbrack\xi_{n1}, 1\rbrack$. Conditions on $\lambda_n$ are given that determine whether $P(A_n \mathrm{i.o.})$ and $P(B_n \mathrm{i.o.})$ equal 0 or 1. Results for $A_n$ (for $B_n$) are related to upper class sequences for $1/(n\xi_{n1})$ (for $n\xi_{n2})$. Upper class sequences for $n\xi_{nk}$, with $k > 1$, are characterized. In the case of nonidentically distributed random variables, we present the result analogous to $P(A_n \mathrm{i.o.}) = 0$.