Approximating Shepp's constants for the Slepian process

Abstract

Slepian process $S(t)$ is a stationary Gaussian process with zero mean and covariance $ES(t)S(t^{\prime })=\max \{0,1-|t-t^{\prime }|\}.$ For any $T>0$ and $h>0$, define $F_T(h)=\mathrm{Pr}\left\{{\max }_{t\in [0,T]}S(t)<h\right\}$ and the constants $\Lambda (h)=-{\lim }_{T\to \infty }\frac{1}{T}\log F_T(h)$ and $\lambda (h)=\exp \{-\Lambda (h)\}$; we will call them `Shepp's constants'. The aim of the paper is construction of accurate approximations for $F_T(h)$ and hence for the Shepp's constants. We demonstrate that at least some of the approximations are extremely accurate.