A limit theorem for expectations conditional on a sum
type: Item
instance of: scholarly article
title: A limit theorem for expectations conditional on a sum
zbMATH Open document ID: 0780.60025
DOI: 10.1007/BF01047574
published in: Journal of Theoretical Probability
publication date: 1994-01-30T00:00:00Z
review text: Let \((\xi_ n)\) be i.i.d. and let \(U\) be another r.v. such that \(U\) and \(\xi_ 1\) may be dependent, but \(U\) is otherwise independent of the sequence. The main result of the paper is that, under appropriate regularity and third moment conditions, \[ \mathbb{E}[U \mid \xi_ 1 + \cdots+\xi_ n=n \mu]=\mathbb{E}[U]+O(n^{-1}), \] and this result is the best possible in the sense that the rate of convergence can be of strictly lesser order if a third moment does not exist. The primary tool employed in the proof of the above equation is a formula for the conditional expectation due to Bartlett; cf. \textit{S. Zabell} [Ann. Probab. 7, 159-165 (1979; Zbl 0392.60013)].
reviewed by: Zbigniew J. Jurek
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 179064
zbMATH Keywords: conditional expectations
zbMATH Keywords: Bartlett's formula
zbMATH Keywords: rate of convergence
author: Sandy L. Zabell
MaRDI profile type: Publication
cites work: The Characteristic Function of a Conditional Statistic
cites work: Q4197800
cites work: Q4114551
cites work: Q5624436
cites work: Q5829358
cites work: Q5796456
cites work: A Local Limit Theorem for Non-Identically Distributed Random Variables
cites work: Asymptotic Form of the Structure Function for Real Systems
cites work: Q4085018
cites work: Continuous versions of regular conditional distributions
cites work: Rates of convergence for conditional expectations
full work available at URL: https://doi.org/10.1007/bf01047574
OpenAlex ID: W2000666821
description: scientific article; zbMATH DE number 179064