  Gaussian errors are sometimes inappropriate in a multivariate linear
regression setting because, for example, the data contain outliers. In such
situations, it is often assumed that the error density is a scale mixture of
multivariate normal densities that takes the form $f(\varepsilon) =
\int_0^\infty |\Sigma|^{-\frac{1}{2}} u^{\frac{d}{2}} \, \phi_d \big(
\Sigma^{-\frac{1}{2}} \sqrt{u} \, \varepsilon \big) \, h(u) \, du$, where $d$
is the dimension of the response, $\phi_d(\cdot)$ is the standard $d$-variate
normal density, $\Sigma$ is an unknown $d \times d$ positive definite scale
matrix, and $h(\cdot)$ is some fixed mixing density. Combining this alternative
regression model with a default prior on the unknown parameters results in a
highly intractable posterior density. Fortunately, there is a simple data
augmentation (DA) algorithm and a corresponding Haar PX-DA algorithm that can
be used to explore this posterior. This paper provides conditions (on $h$) for
geometric ergodicity of the Markov chains underlying these Markov chain Monte
Carlo (MCMC) algorithms. These results are extremely important from a practical
standpoint because geometric ergodicity guarantees the existence of the central
limit theorems that form the basis of all the standard methods of calculating
valid asymptotic standard errors for MCMC-based estimators. The main result is
that, if $h$ converges to 0 at the origin at an appropriate rate, and
$\int_0^\infty u^{\frac{d}{2}} \, h(u) \, du < \infty$, then the DA and Haar
PX-DA Markov chains are both geometrically ergodic. This result is quite
far-reaching. For example, it implies the geometric ergodicity of the DA and
Haar PX-DA Markov chains whenever $h$ is generalized inverse Gaussian,
log-normal, inverted gamma (with shape parameter larger than $d/2$), or
Fr\'{e}chet (with shape parameter larger than $d/2$).
