Inequalities for variances of some functions of random variables
type: Item
instance of: scholarly article
title: Inequalities for variances of some functions of random variables
zbMATH Open document ID: 0827.60012
DOI: 10.1007/BF02926029
author: V. Nollau
published in: Statistical Papers
publication date: 1995-11-29T00:00:00Z
review text: Some inequalities for variances and other measures of deviation of functions of random variables are considered. The following inequality is typical. Let \(f\) be a real-valued function, defined on \((0, \infty)\). If there exists a positive function \(g_f\) on \([0, \infty)\) with the property that \[ \bigl |f(x + h) - f(x) \bigr |\leq |h |/g_f (x) \] for all \(x > 0\) and \(h > - x\), then \[ D^2 \bigl( f(Y) \bigr) \leq D^2Y/ \bigl( g_f (EY) \bigr)^2 \] holds for any nonnegative random variables \(Y\) with \(EY > 0\) and \(EY^2 < \infty\). Based on the inequalities from the paper the author derives some corollaries concerning the variance of powers of random variables and of sums of independent random variables. For example, the following inequality \[ D^2 (X_1 + \cdots + X_n)^\alpha \leq D^2 X_1/n^{1 - 2 \alpha} (EX_1)^{2 - 2 \alpha} \] holds for any independent identically distributed random variables \(X_1, \ldots, X_n\) and \(0 < \alpha \leq 1\).
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 769862
zbMATH Keywords: inequalities for variances
zbMATH Keywords: sums of independent random variables
MaRDI profile type: Publication
cites work: Inequalities for absolute moments of a distribution: From Laplace to von Mises
cites work: Q3757055
cites work: Q3049563
description: scientific article; zbMATH DE number 769862