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Mutual information decay for factors of i.i.d.

Published online by Cambridge University Press:  29 April 2018

BALÁZS GERENCSÉR  and
VIKTOR HARANGI
BALÁZS GERENCSÉR
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Reáltanoda utca 13-15, Hungary email gerencser.balazs@renyi.mta.hu, harangi@renyi.hu ELTE Eötvös Loránd University, Department of Probability and Statistics, H-1117 Budapest, Pázmány Péter sétány 1/c, Hungary
VIKTOR HARANGI
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, H-1053 Budapest, Reáltanoda utca 13-15, Hungary email gerencser.balazs@renyi.mta.hu, harangi@renyi.hu
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Abstract

This paper is concerned with factors of independent and identically distributed processes on the $d$-regular tree for $d\geq 3$. We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 3015 - 3030
Copyright
© Cambridge University Press, 2018 

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