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Extremal probabilities for Gaussian quadratic forms
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  • Published: June 2003

Extremal probabilities for Gaussian quadratic forms

  • Gábor J. Székely1 &
  • Nail K. Bakirov2 

Probability Theory and Related Fields volume 126, pages 184–202 (2003)Cite this article

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Abstract.

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf Q P(Q≤x)=P(χ2 n /n≤x), where χ n 2 is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z 1 ,Z 2 are independent standard Gaussian rv's, then sup0≤λ≤1/2 P{λZ 1 2+(1−λ)Z 2 2≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Bowling Green State University Bowling Green, OH 43403-0221, U.S.A. e-mail: gabors@bgnet.bgsu.edu and Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Budapest, Hungary, , , , , , US

    Gábor J. Székely

  2. Institute of Mathematics, USC Russian Academy of Sciences 112 Chernyshevskii St. 450000 Ufa, Russia. e-mail: bakirov@imat.rb.ru, , , , , , RU

    Nail K. Bakirov

Authors
  1. Gábor J. Székely
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  2. Nail K. Bakirov
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Additional information

Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003

Research supported by NSA Grant MDA904-02-1-0091

Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10

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Székely, G., Bakirov, N. Extremal probabilities for Gaussian quadratic forms. Probab. Theory Relat. Fields 126, 184–202 (2003). https://doi.org/10.1007/s00440-003-0262-6

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  • Issue Date: June 2003

  • DOI: https://doi.org/10.1007/s00440-003-0262-6

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Keywords

  • Similar Statement
  • Quadratic Form
  • Signal Detection
  • Gaussian Process
  • Gaussian Random Variable
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