The Annals of Probability

Rates of Convergence for Conditional Expectations

Sandy L. Zabell

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 8, Number 5 (1980), 928-941.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994622

Digital Object Identifier
doi:10.1214/aop/1176994622

Mathematical Reviews number (MathSciNet)
MR586777

Zentralblatt MATH identifier
0441.60019

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
Conditional expectations rates of convergence ratio limit theorem spacings binomial Poisson convergence equivalence of ensembles

Citation

Zabell, Sandy L. Rates of Convergence for Conditional Expectations. Ann. Probab. 8 (1980), no. 5, 928--941. doi:10.1214/aop/1176994622. https://projecteuclid.org/euclid.aop/1176994622


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