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?