We study a Bayesian linear mixed model whose random effects follow a scale mixture of normal distributions with an arbitrary mixing distribution, a class that contains the Student-t, slash, contaminated normal, normal-generalized inverse Gaussian and Laplace laws. Conditionally conjugate priors lead to a three-block Gibbs sampler that alternates between the variance components, the latent precision weights, and the regression coefficients together with the random effects. Using a drift function built from fractional powers, we give sufficient conditions for geometric ergodicity of this chain, which involve a moment of the mixing distribution of some order larger than 1/2 and its behaviour near the origin. An equivariance of the sampler under rescaling of the design then shows that the conditions hold for all the families above, including Student-t random effects with any degrees of freedom, for every random-effects design matrix, full rank or not, with the rank entering only the conditions on the priors of the variance components. Together with posterior moment bounds, geometric ergodicity justifies Monte Carlo standard errors and effective-sample-size stopping rules for the regression coefficients and the log-variance components. Simulations examine these procedures at practical run lengths: joint confidence regions need a correction for the number of batches, and under sequential stopping their observed coverage remains slightly below the nominal level. We illustrate the methods with longitudinal beta-HCG measurements from early pregnancies, in which the latent weights reveal larger between-subject variability among abnormal pregnancies.

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