# Can a Markov chain be approximated with an AR process?

In some MCMC literature/source code, a Markov chain is often approximated with an AR(1) process. There is some theory to suggest that such an approximation is somewhat valid for a finite state space, but I am not aware of any literature for general state space.

There is a lot of literature out there for approximating an AR process with a Markov chain, but I am interested in the opposite.

Most significantly where I have seen this approximation being used is the coda package in R, where it is used to estimate the spectral density at 0 for a process. Look at the help page for function spectrum0.ar here. This function is then used in the calculation of effective same size, amongst other things.

Seeing the popularity of this package, I am wondering how such an approximation is theoretically valid? Is there literature out there that justifies this approximation?

• @Student001 A step in your logical deduction seems to be broken: (i) AR(1) $\neq$ All Markov Processes, in fact, AR(1) $\subset$ All Markov Processes, (ii) There are some Markov Processes that are from looking like an AR(1), (iii) Even if you have a Markov Process that is AR(1), finding the exact structure is not an easy task (you have to specify/estimate a parameter and the distribution of the "errors" $\epsilon$), that's why you would try to approximate it. – Tea May 20 '16 at 14:57