Stationary gaussian processes with trajectories in generalized Sobolev spaces

Simulating Gaussian Stationary Processes with Unbounded Spectra

We propose a new method for simulating a Gaussian process, whose spectrum diverges at one or multiple frequencies in 0; 1 2 ] (not necessarily at zero). The method utilizes a generalization of the discrete wavelet transform, the discrete wavelet packet transform (DWPT), and only requires explicit knowledge of the spectral density function of the process { not its autocovariance sequence. An ort...

Representations of Gaussian processes with stationary increments

Stationary Systems of Gaussian Processes

We describe all countable particle systems on R which have the following three properties: independence, Gaussianity, and stationarity. More precisely, we consider particles on the real line starting at the points of a Poisson point process with intensity measure m and moving independently of each other according to the law of some Gaussian process ξ. We describe all pairs (m, ξ) generating a s...

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...

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...