Boundary and Entropy of Space Homogeneous Markov Chains

Boundary and Entropy of Space Homogeneous Markov Chains

CNRS, Université de Rennes-1 and Technische Universität Graz We study the Poisson boundary (≡ representation of bounded harmonic functions) of Markov operators on discrete state spaces that are invariant under the action of a transitive group of permutations. This automorphism group is locally compact, but not necessarily discrete or unimodular. The main technical tool is the entropy theory whi...

Taylor Expansion for the Entropy Rate of Hidden Markov Chains

We study the entropy rate of a hidden Markov process, defined by observing the output of a symmetric channel whose input is a first order Markov process. Although this definition is very simple, obtaining the exact amount of entropy rate in calculation is an open problem. We introduce some probability matrices based on Markov chain's and channel's parameters. Then, we try to obtain an estimate ...

The Rate of Rényi Entropy for Irreducible Markov Chains

In this paper, we obtain the Rényi entropy rate for irreducible-aperiodic Markov chains with countable state space, using the theory of countable nonnegative matrices. We also obtain the bound for the rate of Rényi entropy of an irreducible Markov chain. Finally, we show that the bound for the Rényi entropy rate is the Shannon entropy rate.

Transport-entropy inequalities and curvature in discrete-space Markov chains

Let G = (Ω, E) be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Zt} on Ω whose kernel p satisfies p(x, y) > 0 ⇒ {x, y} ∈ E for every x, y ∈ Ω. In words, transitions only occur between neighboring points of the graph. Suppose further that (Ω, p, d) has coarse Ricci curvature at least 1/α in the sense of Ollivier: For all x, y ∈ Ω, it holds that W1(Z1 | {Z0 = x},...

Continuous-time homogeneous Markov chains