Here is a simulation of a one-dimensional "independence sampler" Metropolis algorithm Markov chain, started in stationarity. The target (stationary) distribution is Exp(1), with density exp(-x). The proposal distribution is Exp(k), with density k*exp(-ky). Each proposal is accepted with probability = min(1, exp((k-1)(y-x))).
Try restarting the chain several times to see different simulation runs. Note that the mean sample value "should" be 1.0. However, with the default value of k=5, it is in fact often somewhat less and occasionally much more. (Indeed, it's possible the entire graph will lie above the screen, so you won't see a thing!) This unstable behaviour is typical of non-geometrically ergodic chains (corresponding here to k>1). For k=1, we have perfect i.i.d. sampling; for k<1 the chain is geometrically ergodic. For further discussion of these issues, see the paper Markov chain Monte Carlo: Some practical implications of theoretical results, by G.O. Roberts and J.S. Rosenthal, or my other research papers.
See the chain run!