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Extremal probabilities for Gaussian quadratic forms. (English) Zbl 1031.60018

Denote by \(Q\) an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that \(E(Q)=1\). The main result of the paper is the following statement: \(\inf_QP(Q\leq x)=P(\chi_n^2/n\leq x), x>0\), where \(\chi_n^2\) is chi-square distributed random variable with \(n=n(x)\) degrees of freedom, \(n(x)\) is a non-increasing function of \(x\). Moreover, it is proved that \(n=1\) iff \(x>x(1)\), \(n=2\) iff \(x\in[x(2),x(1)]\), etc, \(n(x)\leq \text{rank}(Q)\), where \(x(1)=1.5364\cdots, x(2)=1.2989\cdots, \ldots\). It is noted that a similar statement is not true for the supremum: if \(1<x<2\) and \(Z_1,Z_2\) are independent standard Gaussian random variables, then \(\sup_{0\leq\lambda\leq 1/2}P\{\lambda Z_1^2+(1-\lambda)Z_2^2\leq x\}\) is taken not at \(\lambda=0\) or at \(\lambda=1/2\) but at \(0<\lambda<\lambda(x)<1/2\), where \(\lambda(x)\) is a continuous, increasing function from \(\lambda(1)=0\) to \(\lambda(2)=1/2\), e.g. \(\lambda(1.5)=.15\cdots\). Applications of the results include asymptotic quantiles of \(U\)- and \(V\)-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.

MSC:

60E15 Inequalities; stochastic orderings
60G15 Gaussian processes
62G10 Nonparametric hypothesis testing
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