Open Access
Translator Disclaimer
October, 1980 Rates of Convergence for Conditional Expectations
Sandy L. Zabell
Ann. Probab. 8(5): 928-941 (October, 1980). DOI: 10.1214/aop/1176994622

Abstract

Let $\{X_n: n \geqslant 1\}$ be a sequence of i.i.d. random variables with bounded continuous density or probability mass function $f(x)$. If $E(\exp(\alpha|X_1|^\beta)) < \infty$ for some $\alpha > 0$ and $0 < \beta \leqslant 1, \mu = L(X_1), c_n = o(n^{1/(2 - \beta)})$ and $h$ is a measurable function such that $M = E(|h(X_1)|\exp(\alpha|X_1|^\beta)) < \infty$, then $$E(h(X_1)|X_1 + \cdots + X_n = n\mu + c_n) = E(h(X_1)) + M\cdot O\big(\frac{1 + |c_n|}{n}\big)$$ uniformly in $h$. It follows that $$\|\mathscr{L} (X_1\mid X_1 + \cdots + X_n = n\mu + c_n) - \mathscr{L}(X_1)\|_{\operatorname{Var}} = O\big(\frac{1 + |c_n|}{n}\big).$$ Applications are given to the binomial-Poisson convergence theorem, spacings, and statistical mechanics.

Citation

Download Citation

Sandy L. Zabell. "Rates of Convergence for Conditional Expectations." Ann. Probab. 8 (5) 928 - 941, October, 1980. https://doi.org/10.1214/aop/1176994622

Information

Published: October, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0441.60019
MathSciNet: MR586777
Digital Object Identifier: 10.1214/aop/1176994622

Subjects:
Primary: 60F05

Rights: Copyright © 1980 Institute of Mathematical Statistics

JOURNAL ARTICLE
14 PAGES


SHARE
Vol.8 • No. 5 • October, 1980
Back to Top