Approximating Shepp's constants for the Slepian process

Abstract

Slepian process $S(t)$ is a stationary Gaussian process with zero mean and covariance $ E S(t)S(t')=\max\{0,1-|t-t'|\}\, . $ For any $T>0$ and $h>0$, define $F_T(h ) = {\rm Pr}\left\{\max_{t \in [0,T]} S(t) < h \right\} $ and the constants $\Lambda(h) = -\lim_{T \to \infty} \frac1T \log F_T(h)$ 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.