Checking MCMC convergence with a single chain I have read the Gelman-Rubin method for check the convergence in MCMC on $m\geq 2$ chain, but when I work with only one chain, what can i do to check the convergence? 
Is there any method that works fine with $m=1$ chain?
 A: First, the Gelman-Rubin test does not check convergence of an MCMC Markov chain but simply an agreement between several parallel chains: if all chains miss a highly concentrated but equally highly important mode of the target distribution, the Gelman-Rubin criterion concludes to the convergence of the chains. Using multiple chains to check for convergence is quite reasonable if costly, but one can never "be sure to have reached stationarity". Simulated tempering can help, though.
Second, to check convergence or stationarity on a single Markov chain $(x_t)_{t=1,\ldots,T}$, one needs to know a lot about the target distribution $\pi(x)$ because, otherwise, all you can judge from the sequence of values $x_1,x_2,\ldots,x_T$ is their stability. Hence only the ability of the MCMC sampler to explore the current region of the support of $\pi$. To go beyond that requires an assessment of this support and of the "missing mass", i.e. the mass under $\pi$ of the remainder of the space. This is an extremely rare occurrence. 
