Metropolis Monte Carlo Autocorrelation

Hello,

I do not quite understand yet how to assess the convergence of a metropolis run using the autocorrelation. I think I get the principle, that if we have less autocorrelation in the Markov chain, the obtained samples are more independent.

For each parameter in my model, I calculate the autocorrelation at time-lag as follows:

where is the mean of .

Like this I calculate the autocorrelations for each parameter for each time lag . So when I do 100 steps in my Metropolis-Algorithm, I calculate all autocorrelations for . Of course, the autocorrelation at lag 0 is 1, and drops to around zero. But in all cases, it drops to around zero, no matter how many steps I do. When I plot the autocorrelation of a MC-Simulation with 1000 steps, then I see that each parameter converges after about 1/4 of the steps. But if I do only 10 steps, I see the same, after about 1/4 of the steps each parameter converges, which cannot be. Where is my mistake in thinking? How can I really use autocorrelation to assess that my ensemble converges?

Many thanks in advance,

Greetings Hans