How to Characterise These Markov Chains? I have the following two Markov chains:
1.

2.

I'm trying to characterise them. Unfortunately, I have no idea how to "characterise" them. At best, I can tell that chain 2 looks a lot "healthier" than chain 1. I've attempted to search for any documents that explain how to characterise Markov chains, but I've been unable to find any. The best I've been able to do is the wikipedia page on Markov chain Monte Carlo and a wikipedia page on stationary processes. However, the former doesn't seem to offer anything useful in this regard, and, since both chains seem to be stationary, nor does the latter. 
Am I correct in saying that Markov chain 2 is "healthier" than Markov chain 1, in that Markov chain 1 seems to be more "jumpy" at some points and less "smooth" overall? How can I characterise these chains? 
I would greatly appreciate it if people could please provide me with some direction, so that I can learn how to "characterise" these Markov chains. I can't seem to find any resources on the matter.
 A: There are two main ways of assessing the quality of a Markov chain


*

*The rate of convergence of the Markov chain to the stationary distribution.  In the literature, the word "mixing" is often used to describe the rate of convergence of the Markov chain. For example, the first Markov chain would be characterized as one that does not mix well, and the second as one which mixes better. Unfortunately, a definitive statement about the mixing usually requires in-depth analysis of the Markov chain transition density.

*The asymptotic variance of sample averages is the second way of assessing the quality of a Markov chain. A sample obtained via a Markov chain has serial correlation, which means that sample averages often have inflated variance. You can find more details here and in the links there-in. Heuristically, one common way the asymptotic variance is assessed is by looking at the autocorrelations in the Markov chain by plotting an ACF plot. Higher number of significant lags in the ACF plot means higher autocorrelations, which would imply larger variance for sample averages. I would guess that the first Markov chain has higher lags in the ACF plot and the second has smaller number of significant lags. Again, plotting the ACF plot only provides a heuristic understanding. To say anything with confidence, you would want to estimate the asymptotic variance directly. You can do this using the R package mcmcse and the mcse function there, as described in the post here. An equivalent way is to compare the effective sample size (ESS) of the two Markov chains. ESS can be estimated with the function ess in the same R package.

