1,688 research outputs found
Two-sided shift spaces over infinite alphabets
Ott, Tomforde, and Willis proposed a useful compactification for one-sided
shifts over infinite alphabets. Building from their idea we develop a notion of
two-sided shift spaces over infinite alphabets, with an eye towards
generalizing a result of Kitchens. As with the one-sided shifts over infinite
alphabets our shift spaces are compact Hausdorff spaces but, in contrast to the
one-sided setting, our shift map is continuous everywhere. We show that many of
the classical results from symbolic dynamics are still true for our two-sided
shift spaces. In particular, while for one-sided shifts the problem about
whether or not any -step shift is conjugate to an edge shift space is open,
for two-sided shifts we can give a positive answer for this question.Comment: 32 page
Consistency of Feature Markov Processes
We are studying long term sequence prediction (forecasting). We approach this
by investigating criteria for choosing a compact useful state representation.
The state is supposed to summarize useful information from the history. We want
a method that is asymptotically consistent in the sense it will provably
eventually only choose between alternatives that satisfy an optimality property
related to the used criterion. We extend our work to the case where there is
side information that one can take advantage of and, furthermore, we briefly
discuss the active setting where an agent takes actions to achieve desirable
outcomes.Comment: 16 LaTeX page
A Generalized Typicality for Abstract Alphabets
A new notion of typicality for arbitrary probability measures on standard
Borel spaces is proposed, which encompasses the classical notions of weak and
strong typicality as special cases. Useful lemmas about strong typical sets,
including conditional typicality lemma, joint typicality lemma, and packing and
covering lemmas, which are fundamental tools for deriving many inner bounds of
various multi-terminal coding problems, are obtained in terms of the proposed
notion. This enables us to directly generalize lots of results on finite
alphabet problems to general problems involving abstract alphabets, without any
complicated additional arguments. For instance, quantization procedure is no
longer necessary to achieve such generalizations. Another fundamental lemma,
Markov lemma, is also obtained but its scope of application is quite limited
compared to others. Yet, an alternative theory of typical sets for Gaussian
measures, free from this limitation, is also developed. Some remarks on a
possibility to generalize the proposed notion for sources with memory are also
given.Comment: 44 pages; submitted to IEEE Transactions on Information Theor
Attractive regular stochastic chains: perfect simulation and phase transition
We prove that uniqueness of the stationary chain, or equivalently, of the
-measure, compatible with an attractive regular probability kernel is
equivalent to either one of the following two assertions for this chain: (1) it
is a finitary coding of an i.i.d. process with countable alphabet, (2) the
concentration of measure holds at exponential rate. We show in particular that
if a stationary chain is uniquely defined by a kernel that is continuous and
attractive, then this chain can be sampled using a coupling-from-the-past
algorithm. For the original Bramson-Kalikow model we further prove that there
exists a unique compatible chain if and only if the chain is a finitary coding
of a finite alphabet i.i.d. process. Finally, we obtain some partial results on
conditions for phase transition for general chains of infinite order.Comment: 22 pages, 1 pseudo-algorithm, 1 figure. Minor changes in the
presentation. Lemma 6 has been remove
Phase transitions for suspension flows
This paper is devoted to study thermodynamic formalism for suspension flows
defined over countable alphabets. We are mostly interested in the regularity
properties of the pressure function. We establish conditions for the pressure
function to be real analytic or to exhibit a phase transition. We also
construct an example of a potential for which the pressure has countably many
phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note
"Thermodynamic formalism for the positive geodesic flow on the modular
surface" arXiv:1009.462
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