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Spectral subsampling MCMC was recently proposed to speed up Markov chainMonte Carlo (MCMC) for long stationary univariate time series by subsamplingperiodogram observations in the frequency domain. This article extends theapproach to multivariate time series using a multivariate generalisation of theWhittle likelihood. To assess the computational gains from spectral subsamplingin challenging problems, a multivariate generalisation of the autoregressivetempered fractionally integrated moving average model (ARTFIMA) is introducedand some of its properties derived. Bayesian inference based on the Whittlelikelihood is demonstrated to be a fast and accurate alternative to the exacttime domain likelihood. Spectral subsampling is shown to provide up to twoorders of magnitude additional speed-up, while retaining MCMC samplingefficiency and accuracy, compared to spectral methods using the full dataset.  Keywords: Bayesian, Markov chain Monte Carlo, Semi-long memory, Spectralanalysis, Whittle likelihood.

Spectral Subsampling MCMC for Stationary Multivariate Time Series with Applications to Vector ARTFIMA Processes