For a $2\times2$ markov chain, with $p$ the probability to remain in state 1, and $q$ the probability to move to state $1$ from state $2$,
I know I can also completely describe the markov chain with
- its autocorrelation $r=p-q$ (which is also my second eigenvalue)
- and with the state $1$ asymptotic frequency $f_1=\dfrac{q}{1-r}$.
I would like to do the same with a $3\times3$ markov chain (therefore with $6$ degrees of freedom), using $f_1$ and $f_2$ (the frequencies of the first $2$ states), the autocorrelation $r$, and (obviously) $3$ other parameters (but which ?)
I guess this may be related to a more general question about the relationship between the autocorrelation coefficient, the asymptotic frequencies and the eigenvalues of a $3\times3$ markov chain.
Thanks for your help