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We investigate a class of estimators of the Markov order for stationaryergodic processes which form a slight modification of the constructions byMerhav, Gutman, and Ziv in 1989 as well as by Ryabko, Astola, and Malyutov in2006 and 2016. All the considered estimators compare the estimate of theentropy rate given by a universal code with the empirical conditional entropyof a string and return the order for which the two quantities are approximatelyequal. However, our modification, which we call universal Markov orders,satisfies a few attractive properties, not shown by the mentioned authors fortheir original constructions. Firstly, the universal Markov orders are almostsurely consistent, without any restrictions. Secondly, they are upper boundedasymptotically by the logarithm of the string length divided by the entropyrate. Thirdly, if we choose the Prediction by Partial Matching (PPM) as theuniversal code then the number of distinct substrings of the length equal tothe universal Markov order constitutes an upper bound for the block mutualinformation. Thus universal Markov orders can be also used indirectly forquantification of long memory for an ergodic process.

On a Class of Markov Order Estimators Based on PPM and Other Universal Codes