Geometric ergodicity of Gibbs sampler for Bayesian linear regression with tail adaptive shrinkage

We study the GLT prior introduced by Lee et al. (2024), a global–local–tail shrinkage prior developed for high-dimensional linear regression, and establish rigorous convergence guarantees for the associated Gibbs sampler developed in Section A.1.1. in Lee et al. (2024). The GLT prior extends classical global–local shrinkage formulations by introducing an explicit tail-index parameter, thereby enabling simultaneous control of near-zero regularization and heavy-tailed robustness. This flexibility permits aggressive shrinkage of noise-dominated coefficients while allowing large signals to escape attenuation through automatic, data-adaptive adjustment to the sparsity level, making the prior well-suited for diverse sparsity regimes. Posterior computation is carried out via a Gibbs sampler that combines conjugate updates for the regression coefficients and noise variance with slice-sampling and elliptical slice-sampling updates for the hierarchy of local, global, and tail parameters. We derive explicit Foster–Lyapunov drift conditions together with a one-step minorization condition for the full transition kernel of the Gibbs sampler, and show that the resulting Markov chain is ψ-irreducible, aperiodic, Harris recurrent, and geometrically ergodic, under mild assumptions. Geometric ergodicity in turn yields a central limit theorem for ergodic averages, thereby providing a rigorous theoretical foundation for reliable MCMC-based inference under high-dimensional linear regression with the GLT prior.