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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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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DOI: https://doi.org/10.1007/s00440-003-0262-6
Keywords
- Similar Statement
- Quadratic Form
- Signal Detection
- Gaussian Process
- Gaussian Random Variable