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Convergence properties of the Gibbs sampler for perturbations of Gaussians. (English) Zbl 0854.60066

Summary: The exact second eigenvalue of the Markov operator of the Gibbs sampler with random sweep strategy for Gaussian densities is calculated. A comparison lemma yields an upper bound on the second eigenvalue for bounded perturbations of Gaussians which is a significant improvement over previous bounds. For two-block Gibbs sampler algorithms with a perturbation of the form \(\chi (g_1 (x^{(1)}) + g_2 (x^{(2)}))\) the derivative of the second eigenvalue of the algorithm is calculated exactly at \(\chi = 0\), in terms of expectations of the Hessian matrices of \(g_1\) and \(g_2\).

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
47B38 Linear operators on function spaces (general)
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