Order-of-magnitude bounds for expectations involving quadratic forms
type: Item
instance of: scholarly article
title: Order-of-magnitude bounds for expectations involving quadratic forms
zbMATH Open document ID: 0810.60012
author: Victor H. de la Peña
author: Michael J. Klass
published in: The Annals of Probability
publication date: 1995-04-04T00:00:00Z
review text: Let \(X = (X_ 1, \dots, X_ n)\) be \(n\) independent r.v.s, \((a_{ij}) \in \mathbb{R}^{n \times n}\) a symmetric matrix with \(a_{jj} = 0\), and \(\Phi\) be a positive symmetric function s.t. \(\Phi (0) = 0\). If \(\Phi (2x) \leq 2^ \beta \Phi (x)\) holds and if \(\Phi (x)\) is convex and \(\Phi (\sqrt x)\) concave, the authors obtain upper and lower bounds for \(\mathbb{E} (\Phi (\sum^ n_{ij} a_{ij} X_ i X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Phi (\sum^ m_{ij} a_{ij} X_ i X_ j))\) giving the order of magnitude of the quadratic form. This result can be generalized to the case where for a \(k \in \mathbb{N}_ 0\) both \(\Phi (x^{2 - k})\) is convex and \(\Phi (x^{2 - k - 1})\) is concave. The proofs rely on a decoupling technique stating that for arbitrary convex functions \(\Psi\) the expressions \(\mathbb{E} (\Psi (\sum^ n_{ij} a_{ij} X_ i X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Psi (\sum^ m_{ij} a_{ij} X_ i X_ j))\) can be compared from above and below with \(\mathbb{E} (\Psi (\sum^ n_{ij} a_{ij} X_ i \widetilde X_ j))\) and \(\mathbb{E} (\max_{m \leq n} \Psi (\sum^ m_{ij} a_{ij} X_ i \widetilde X_ j))\), respectively. Here, \(\widetilde X\) denotes an independent copy of \(X\). This technique also yields a version of Khinchin's inequality, namely \[ \mathbb{E} \Bigl( \sum^ n_{ij} a_{ij} X_ i X_ j \Bigr) \sim \mathbb{E} \biggl( \Bigl( \sum^ n_{ij} a^ 2_{ij} X^ 2_ iX^ 2_ j \Bigr)^{1/2} \biggr) \sim \mathbb{E} \biggl( \Bigl( \sum^ n_{ij} a^ 2_{ij} X ^ 2_ i \widetilde X^ 2_ j \Bigr)^{1/2} \biggl). \]
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60F25
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 681393
zbMATH Keywords: quadratic forms of random variables
zbMATH Keywords: decoupling inequalities
zbMATH Keywords: Khinchin's inequality
zbMATH Keywords: maximal inequalities
zbMATH Keywords: decoupling technique
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988740
OpenAlex ID: W2068675397
DOI: 10.1214/AOP/1176988740
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description: scientific article; zbMATH DE number 681393