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April, 1981 The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers
F. T. Wright
Ann. Probab. 9(2): 323-329 (April, 1981). DOI: 10.1214/aop/1176994475

Abstract

Let $\{X_k\}$ be a sequence of independent random variables which are centered at their means; let $\{T_k\}$ be an i.i.d. sequence of $\beta$-dimensional random vectors with common distribution $\mu$; and let $\{X_k\}$ and $\{T_k\}$ be independent. With $\mathscr{L}$ the collection of lower layers, a necessary and sufficient condition for the almost sure convergence of $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} \chi L (T_k)/n - \mu(L)|$ to zero is given. In addition, this condition on $\mu$ is shown to imply that $\sup_{L \in \mathscr{L}}|\sum^n_{k = 1} X_{k\chi L}(T_k)|/n \rightarrow 0$ a.s. provided the $X_k$ satisfy a first moment-like condition. Rates of convergence are also investigated.

Citation

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F. T. Wright. "The Empirical Discrepancy Over Lower Layers and a Related Law of Large Numbers." Ann. Probab. 9 (2) 323 - 329, April, 1981. https://doi.org/10.1214/aop/1176994475

Information

Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60032
MathSciNet: MR606996
Digital Object Identifier: 10.1214/aop/1176994475

Subjects:
Primary: 60F15
Secondary: 62G05

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 2 • April, 1981
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