Order estimation of Markov chains

TL;DR

This paper introduces estimators that determine the order of a stationary Markov chain from data, converging almost surely to the true order or infinity if the process is not Markovian.

Contribution

The paper presents a new estimator for the order of Markov chains that guarantees almost sure convergence to the true order or infinity.

Findings

Estimator converges almost surely to the true Markov order.

Estimator distinguishes Markov chains from non-Markov processes.

Method applicable to processes over countable alphabets.

Abstract

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