On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

Realization Theory for Multivariate Stationary Gaussian Processes

This paper collects in one place a comprehensive theory of stochastic realization for continuous time stationary Gaussian vector processes which in various pieces has appeared in a number of our earlier papers. It begins with an abstract state space theory, based on the concept of splitting subspace. These results are then carried over to the spectral domain and described in terms of Hardy func...

Small Deviations of Smooth Stationary Gaussian Processes

We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson’s upper b...

The Intersection of past and Future for Multivariate Stationary Processes

We consider an intersection of past and future property of multivariate stationary processes which is the key to deriving various representation theorems for their linear predictor coefficient matrices. We extend useful spectral characterizations for this property from univariate processes to multivariate processes.

Piecewise Stationary Markov Decision Processes, II: State-Dependent Gain

Completely Regular Multivariate Stationary Processes and the Muckenhoupt Condition

In this paper we shall give a necessary and sufficient condition for a multivariate stationary stochastic process to be completely regular. For the scalar case the description of completely regular processes was obtained by Helson an Sarason, see [2, 9]. Almost none of the scalar methods is available in the vector situation. The explanation is simple. Our problem will be reduced to verifying L2...