  While there is an increasing amount of literature about Bayesian time series
analysis, only a few Bayesian nonparametric approaches to multivariate time
series exist. Most methods rely on Whittle's Likelihood, involving the second
order structure of a stationary time series by means of its spectral density
matrix. This is often modeled in terms of the Cholesky decomposition to ensure
positive definiteness. However, asymptotic properties such as posterior
consistency or posterior contraction rates are not known. A different idea is
to model the spectral density matrix by means of random measures. This is in
line with existing approaches for the univariate case, where the normalized
spectral density is modeled similar to a probability density, e.g. with a
Dirichlet process mixture of Beta densities. In this work, we present a related
approach for multivariate time series, with matrix-valued mixture weights
induced by a Hermitian positive definite Gamma process. The proposed procedure
is shown to perform well for both simulated and real data. Posterior
consistency and contraction rates are also established.
