Small ball probabilities for the Slepian Gaussian fields

Small Ball Probabilities for the Slepian Gaussian Fields

The d-dimensional Slepian Gaussian random field {S(t), t ∈ R+} is a mean zero Gaussian process with covariance function ES(s)S(t) = ∏d i=1 max(0, ai − |si − ti|) for ai > 0 and t = (t1, · · · , td) ∈ R+. Small ball probabilities for S(t) are obtained under the L2-norm on [0, 1]d, and under the sup-norm on [0, 1]2 which implies Talagrand’s result for the Brownian sheet. The method of proof for t...

Small Ball Probabilities for Gaussian Markov

Gaussian Processes: Inequalities, Small Ball Probabilities and Applications

Small ball probabilities for Gaussian Markov processes under the Lp - norm ( Wenbo

Let {X (t); 06t61} be a real-valued continuous Gaussian Markov process with mean zero and covariance (s; t)=EX (s)X (t) 6= 0 for 0¡s; t ¡ 1. It is known that we can write (s; t)= G(min(s; t))H (max(s; t)) with G¿ 0; H ¿ 0 and G=H nondecreasing on the interval (0; 1). We show that for the Lp-norm on C[0; 1], 16p6∞ lim →0 2 logP(‖X (t)‖p ¡ ) =− p (∫ 1 0 (G′H − H ′G)p=(2+p) dt )(2+p)=p and its var...

Small-Ball Probabilities for the Volume of Random Convex Sets

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.