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Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
The Evolution of a Spatial Stochastic Network
The asymptotic behavior of a stochastic network represented by a birth and
death processes of particles on a compact state space is analyzed. Births:
Particles are created at rate and their location is independent of
the current configuration. Deaths are due to negative particles arriving at
rate . The death of a particle occurs when a negative particle
arrives in its neighborhood and kills it. Several killing schemes are
considered. The arriving locations of positive and negative particles are
assumed to have the same distribution. By using a combination of monotonicity
properties and invariance relations it is shown that the configurations of
particles converge in distribution for several models. The problems of
uniqueness of invariant measures and of the existence of accumulation points
for the limiting configurations are also investigated. It is shown for several
natural models that if then the asymptotic configuration
has a finite number of points with probability 1. Examples with
and an infinite number of particles in the limit are also
presented
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