Extremal probabilities for Gaussian quadratic forms

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, infQP(Q≤x)=P(χ2n/n≤x), where χn2 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...

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Bibliographic Details
Published in:Probability theory and related fields 2003-06, Vol.126 (2), p.184-202
Main Authors: SZEKELY, Gabor J, BAKIROV, Nail K
Format: Article
Language:English
Subjects:
Continuity (mathematics)
Distribution theory
Exact sciences and technology
Gaussian process
Mathematics
Probability
Probability and statistics
Probability theory and stochastic processes
Quadratic forms
Quantiles
Random variables
Sciences and techniques of general use
Signal detection
Signal processing
Stochastic analysis
Stochastic processes
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Summary: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, infQP(Q≤x)=P(χ2n/n≤x), where χn2 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
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-003-0262-6