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′)=max{0,1−|t−t′|}. For any T≥0 and real h, define FT(h)=Prmaxt∈[0,T]S(t)<h and the constants Λ(h)=−limT→∞1TlogFT(h) and λ(h)=exp{−Λ(h)}; we will call them ‘Shepp’s constants’. The aim of the paper is construction of accurate approximations for FT(h) and hence for the Shepp’s constants. We demonstrate that at least some of the approximations are extremely accurate.