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August 2021 Variable Length Memory Chains: Characterization of stationary probability measures
Peggy Cénac, Brigitte Chauvin, Camille Noûs, Frédéric Paccaut, Nicolas Pouyanne
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Bernoulli 27(3): 2011-2039 (August 2021). DOI: 10.3150/20-BEJ1299


Variable Length Memory Chains (VLMC), which are generalizations of finite order Markov chains, are an essential tool to modelize random sequences in many domains, as well as an interesting object in contemporary probability theory. The question of existence of stationary probability measures leads us to introduce a key combinatorial structure for words produced by a VLMC: the Longest Internal Suffix. This notion allows us to state a necessary and sufficient condition for a VLMC to admit a unique invariant probability measure.

This condition turns out to get a much simpler form for a subclass of VLMC: the stable VLMC. This natural subclass, unlike the general case, enjoys a renewal property. Namely, a stable VLMC induces a semi-Markov chain on an at most countable state space. Unfortunately, this discrete time renewal process does not contain the whole information of the VLMC, preventing the study of a stable VLMC to be reduced to the study of its induced semi-Markov chain. For a subclass of stable VLMC, the convergence in distribution of a VLMC towards its stationary probability measure is established.

Finally, finite state space semi-Markov chains turn out to be very special stable VLMC, shedding some new light on their limit distributions.


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Peggy Cénac. Brigitte Chauvin. Camille Noûs. Frédéric Paccaut. Nicolas Pouyanne. "Variable Length Memory Chains: Characterization of stationary probability measures." Bernoulli 27 (3) 2011 - 2039, August 2021.


Received: 1 April 2020; Revised: 1 November 2020; Published: August 2021
First available in Project Euclid: 10 May 2021

Digital Object Identifier: 10.3150/20-BEJ1299

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 3 • August 2021
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