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Lumpings of Markov Chains, Entropy Rate Preservation, and Higher-Order Lumpability

  • Bernhard C. Geiger (a1) and Christoph Temmel (a2)

A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

Corresponding author
Postal address: Institute for Communications Engineering, Technische Universiät München, Theresienstrasse 90, 80333 Munich. Email address:
∗∗ Postal address: Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands. Email address:
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