Renewal Theory for Sampling Without Replacement

Abstract

Let $\pi$ be a finite set, $\lambda$ a probability measure on $\pi, 0 < x < 1$ and $a \in \pi$. Let $P(a, x)$ denote the probability that in a random order of $\pi, a$ is the first element (in the order) for which the $\lambda$-accumulated sum exceeds $x$. The main result of the paper is that for every $\varepsilon > 0$ there exist constants $\delta > 0$ and $K > 0$ such that if $\rho = \max_{a\in\pi} \lambda(a) < \delta$ and $\mathrm{K}\rho < x < 1 - \mathrm{K}\rho$ then $\sum_{a\in\pi} |P(a, x) - \lambda(a)| < \varepsilon$. This result implies a new variant of the classical renewal theorem, in which the convergence is uniform on classes of random variables.