On small deviations of stationary Gaussian processes and related analytic inequalities

Let {Xj , j ∈ Z} be a Gaussian stationary sequence having a spectral function F of infinite type. Then for all n and z ≥ 0, P { n sup j=1 |Xj | ≤ z } ≤ (∫ z/√G(f) −z/ √ G(f) e−x /2 dx √ 2π )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 ∈ 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), P { sup 0≤s,t≤T |X(s)−X(t)| ≤ a } ≤ exp { − KT ε(a)p(ε(a)) } , where ε(a) = min { b > 0 : δ(b) ≥ a}, δ(b) = minu≥1{√2(1− γ((ub)), u ≥ 1}, and p(b) = 1 + ∑∞j=2 |2γ(jb) − γ((j − 1)b)− γ((j + 1)b)|/2(1 − γ(b)). The proof is based on some decoupling inequalities arising from BrascampLieb 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. AMS (2000) subject classification. Primary 60F15, 60G10, 60G22, 60G50; Secondary 60F05, 11C20, 15B05, 15A42, 34L15.