Stationary Tangent: the Discrete and Non-smooth Case

In [5] we define a stationary tangent process, or a locally optimal stationary approximation, to a real non-stationary smooth Gaussian process. This paper extends the idea by constructing a discrete tangent – a “locally” optimal stationary approximation – for a discrete time, real Gaussian process. Analogously to the smooth case, our construction relies on a generalization of the recursion formula for the orthogonal polynomials of the spectral distribution function. More precisely, we use a generalization of the Schur parameters to identify the stationary tangent. By way of discretizing, we later demonstrate how this tangent can be used to obtain “good” local stationary approximations to non-smooth continuous time, real Gaussian processes. Further, we demonstrate how, analogously to the curvatures in the smooth case, the Schur parameters can be used to determine the order of stationarity of a non-smooth process.