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October, 1973 Convergence to Zero of Quadratic Forms in Independent Random Variables
Gary N. Griffiths, Ronald D. Platt, F. T. Wright
Ann. Probab. 1(5): 838-848 (October, 1973). DOI: 10.1214/aop/1176996849

Abstract

Let $\{X_n\}$ be a sequence of independent random variables none of which are degenerate and define for $y \geqq 0, F(y) = \sup_k P\lbrack |X_k| \geqq y \rbrack$ and $G(y) = \sup_{j \neq k} P\lbrack |X_j X_k| \geqq y \rbrack$. Relationships between the rate of convergence of $F$ and $G$ to zero are investigated. Set $Q_N = \sum_{j, k}a_{jk,N} X_jX_k$ for $N = 1, 2, \cdots$. If the $X$'s are symmetric then it is shown that $Q_N$ converges to zero in probability for a large class of weights $\{a_{jk,N}\}$ if and only if $\lim_{y\rightarrow\infty} yG(y) = 0$. Convergence results are also given for the case when the random variables are not symmetric.

Citation

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Gary N. Griffiths. Ronald D. Platt. F. T. Wright. "Convergence to Zero of Quadratic Forms in Independent Random Variables." Ann. Probab. 1 (5) 838 - 848, October, 1973. https://doi.org/10.1214/aop/1176996849

Information

Published: October, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0272.60015
MathSciNet: MR358925
Digital Object Identifier: 10.1214/aop/1176996849

Subjects:
Primary: 60F05
Secondary: 60F15

Keywords: Almost sure convergence , products of independent random variables , Quadratic forms , tail probabilities , weak convergence

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 5 • October, 1973
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