I am interested in the convergence properties of the Metropolis-within-Gibbs sampler with Independent or Random walk. In this paper, I have read that in the case of an Independent walk, the proposal distribution must satisfy an inequality of the form $p(z) \geq \varepsilon \pi(z)$ (where $p$ denotes the proposal distribution and $\pi$ the target distribution) to ensure that the convergence of the sampler to the stationary distribution is uniformly ergodic. If the proposal distribution does not satisfy this condition, the sampler can have horrible convergence properties.
However, in the paper mentioned above, the condition is given for an adaptive Independent Metropolis-Hastings algorithm. Do you know of a reference which states this result for a classical Independent Metropolis-Hastings algorithm ? More precisely, how horrible can be the convergence of the sampler if the proposal does not satisfy this inequality ? And lastly : is there a similar condition for the Random walk Metropolis-Hastings algorithm ?