Order estimation of Markov chains

We describe estimators ${\chi }_n(X_0,X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$, which when applied to an unknown stationary process taking values from a countable alphabet $\mathcal{X}$, converge almost surely to $k$ in case the process is a $k$-th order Markov chain and to infinity otherwise.