  We establish conditions under which Metropolis-Hastings (MH) algorithms with
a position-dependent proposal covariance matrix will or will not have the
geometric rate of convergence. Some of the diffusions based MH algorithms like
the Metropolis adjusted Langevin algorithm (MALA) and the pre-conditioned MALA
(PCMALA) have a position-independent proposal variance. Whereas, for other
modern variants of MALA like the manifold MALA (MMALA) that adapt to the
geometry of the target distributions, the proposal covariance matrix changes in
every iteration. Thus, we provide conditions for geometric ergodicity of
different variations of the Langevin algorithms. These results have important
practical implications as these provide crucial justification for the use of
asymptotically valid Monte Carlo standard errors for Markov chain based
estimates. The general conditions are verified in the context of conditional
simulation from the two most popular generalized linear mixed models (GLMMs),
namely the binomial GLMM with the logit link and the Poisson GLMM with the log
link. Empirical comparison in the framework of some spatial GLMMs shows that
the computationally less expensive PCMALA with an appropriately chosen
pre-conditioning matrix may outperform the MMALA.
