期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2013
卷号：75
期号：2
页码：139-170
DOI：10.1007/s13171-013-0036-x
语种：English
出版社：Indian Statistical Institute
摘要：Let \( \{ X_j, j\in {\mathbb Z}\}\) be a Gaussian stationary sequence having a spectral function F of infinite type. Then for all n and z ≥ 0, $$ {\mathbb P}\left\{ \sup_{j=1}^n |X_j|\le z \right\}\le \Bigg( \int_{-z/\sqrt{G(f)}}^{z/\sqrt{G(f)}} e^{-x^2/2}\frac{{\rm d} x}{\sqrt{2\pi}} \Bigg)^n, $$ where G ( f ) is the geometric mean of the Radon Nykodim derivative of the absolutely continuous part f of F . The proof uses properties of finite Toeplitz forms. Let \( \{X(t), t\in {\mathbb R}\}\) be a sample continuous stationary Gaussian process with covariance function γ > 0. We also show that there exists an absolute constant K such that for all T > 0, a > 0 with T ≥ ε ( a ), $$ {\mathbb P} \bigg\{ \sup_{0\le s,t\le T} |X(s)-X(t)|\le a\bigg\} \le \exp \Bigg\{-{ KT \over {\varepsilon}(a) p({\varepsilon}(a))}\Bigg\}, $$ where \({\varepsilon} (a)= \min\big\{ b>0: \delta (b)\ge a\big\}\) , \(\delta (b)=\min_{u\ge 1}\{ \sqrt{2(1-{\gamma}((ub))}, u\ge 1\}\) , and \(p(b) = 1+\sum_{j=2}^\infty {|2{\gamma} (jb )-{\gamma} ((j-1)b )-{\gamma} ((j+1)b )|}/{2(1-{\gamma}(b))}\) . The proof is based on some decoupling inequalities arising from Brascamp-Lieb inequality, from which we also derive a general upper bound for the small values of stationary sample continuous Gaussian processes. A two-sided inequality for correlated suprema in the case of the Ornstein-Uhlenbeck process. We also establish an unexpected link between the Littlewood hypothesis and small values of cyclic Gaussian processes. In the discrete case, we obtain a general bound by combining Anderson’s inequality with a weighted inequality for quadratic forms. In doing so, we also clarify link between matrices with dominant principal diagonal and Geršgorin’s disks. Both approaches are developed and compared on examples.
关键词：Small deviation ; Gaussian process ; stationary ; decoupling coefficient ; matrices with dominant principal diagonal ; Geršgorin’s disks ; Toeplitz forms ; eigenvalues ; strong Szegö limit theorem ; geometric mean ; Littlewood hypothesis
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