A possible definition of a stationary tangent

Abstract

This paper offers a way to construct a locally optimal stationary approximation for a non-stationary Gaussian process. In cases where this construction leads to a unique stationary approximation we call it a stationary tangent. This is the case with Gaussian processes governed by smooth n-dimensional correlations. We associate these correlations with equivalence classes of curves in R n . These are described in terms of “curvatures” (closely related to the classical curvature functions); they are constant if and only if the correlation is stationary. Thus, the stationary tangent, at t= t 0, to a smooth correlation, curve or process, is the one with the same curvatures at t 0 (but constant). We show that the curvatures measure the quality of a local stationary approximation and that the tangent is optimal in this regard. These results extend to the smooth infinite-dimensional case although, since the equivalence between correlations and curvatures breaks down in the infinite-dimensional setting, we cannot, in general, single out a unique tangent. The question of existence and uniqueness of a stationary process with given curvatures is intimately related with the classical moment problem and is studied here by using tools from operator theory. In particular, we find that there always exists an optimal Gaussian approximation (defined via the curvatures). Finally, by way of discretizing we introduce the notion of δ-curvatures designed to address non-smooth correlations.