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May, 1982 Renewal Theory for Sampling Without Replacement
Abraham Neyman
Ann. Probab. 10(2): 464-481 (May, 1982). DOI: 10.1214/aop/1176993870


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.


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Abraham Neyman. "Renewal Theory for Sampling Without Replacement." Ann. Probab. 10 (2) 464 - 481, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0485.60083
MathSciNet: MR647517
Digital Object Identifier: 10.1214/aop/1176993870

Primary: 60K70
Secondary: 60G30 , 90E70

Keywords: renewal theory , sampling without replacement

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • May, 1982
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