Renewal Theory for Sampling Without Replacement
Abraham Neyman · The Annals of Probability · 1982
Let $\pi$ be a finite set, $\lambda$ a probability measure on $\pi, 0 0$ 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.