On small deviations of stationary Gaussian processes and related analytic inequalities

Abstract

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.