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

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 Gaussian Markov

Small Deviations for Gaussian Markov Processes Under the Sup-Norm

\. INTRODUCTION Let {X(t); 0 < t < 1} be a real-valued mean-zero Gaussian process, and let "||.||" be a semi-norm on the space of real functions on [0, 1]. The so called "small ball estimates" or "small deviation estimates" refer to the asymptotic behavior of This type of problem is quite delicate, and the asymptotic decay rate in (1.1), up to a constant, depends heavily on the process X(t) and...

Gaussian Processes: Inequalities, Small Ball Probabilities and Applications

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...