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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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