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Let $\pi$ denote the intractable posterior density that results when thelikelihood from a multivariate linear regression model with errors from a scalemixture of normals is combined with the standard non-informative prior. Thereis a simple data augmentation algorithm (based on latent data from the mixingdensity) that can be used to explore $\pi$. Let $h(\cdot)$ and $d$ denote themixing density and the dimension of the regression model, respectively. Hobertet al. (2016) [arXiv:1506.03113v2] have recently shown that, if $h$ convergesto 0 at the origin at an appropriate rate, and $\int_0^\infty u^{\frac{d}{2}}\, h(u) \, du < \infty$, then the Markov chains underlying the DA algorithm andan alternative Haar PX-DA algorithm are both geometrically ergodic. In fact,something much stronger than geometric ergodicity often holds. Indeed, it isshown in this paper that, under simple conditions on $h$, the Markov operatorsdefined by the DA and Haar PX-DA Markov chains are trace-class, i.e., compactwith summable eigenvalues. Many of the mixing densities that satisfy Hobert etal.'s (2016) conditions also satisfy the new conditions developed in thispaper. Thus, for this set of mixing densities, the new results provide asubstantial strengthening of Hobert et al.'s (2016) conclusion without anyadditional assumptions. For example, Hobert et al. (2016) showed that the DAand Haar PX-DA Markov chains are geometrically ergodic whenever the mixingdensity is generalized inverse Gaussian, log-normal, Fr\'{e}chet (with shapeparameter larger than $d/2$), or inverted gamma (with shape parameter largerthan $d/2$). The results in this paper show that, in each of these cases, theDA and Haar PX-DA Markov operators are, in fact, trace-class.

Trace-class Monte Carlo Markov Chains for Bayesian Multivariate Linear Regression with Non-Gaussian Errors