Countable Alphabet Research Articles

Perfect simulation for locally continuous chains of infinite order

We establish sufficient conditions for perfect simulation of chains of infinite order on a countable alphabet. The new assumption, localized continuity, is formalized with the help of the notion of context trees, and includes the traditional continuous case, probabilistic context trees and discontinuous kernels. Since our assumptions are more refined than uniform continuity, our algorithms perfectly simulate continuous chains faster than the existing algorithms of the literature. We provide several illustrative examples.

Attractive regular stochastic chains: perfect simulation and phase transition

AbstractWe prove that uniqueness of the stationary chain, or equivalently, of the$g$-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 independent and identically distributed (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.

The distribution of the short-return function

We consider the first return of a sequence to itself . It is known that Tn/n converges to 1 almost surely for ergodic processes with positive entropy. Large deviations are also known. We study the fluctuations of Tn. For product measures over a countable alphabet we prove the convergence in distribution of Tn, properly re-scaled, to a non-degenerated, non-parametric, distribution. We also show that the re-scaled factors behave as follows: expectation of Tn is about n minus a positive constant and the variance is about a constant. As an application, we compute bounds and limiting values of the proportion of self-avoiding sequences. We illustrate with some examples.

Phase Transitions for Suspension Flows

This paper is devoted to studying the 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.

Expanding maps, shrinking targets and hitting times

We study the (metric) Diophantine approximation properties of uniformly expanding transformations and some non-uniformly expanding transformations, i.e. transformations T(x) with an associated countable (not necessarily finite) partition and a return time function R(x) (constant on the blocks of the partition) so that is uniformly expanding, and we obtain Borel–Cantelli results on hitting times of shrinking targets. Our arguments do not require the so-called big image property for and our results contain most of the diversity of examples of slowly mixing systems. We also obtain, with related techniques, results for one-sided topological Markov chains over a countable alphabet with a Gibbs measure.

A Normal Law for the Plug-in Estimator of Entropy

This paper establishes a sufficient condition for the asymptotic normality of the plug-in estimator of Shannon's entropy defined on a countable alphabet. The sufficient condition covers a range of cases with countably infinite alphabets, for which no normality results were previously known.

Entropy Estimation in Turing's Perspective

A new nonparametric estimator of Shannon's entropy on a countable alphabet is proposed and analyzed against the well-known plug-in estimator. The proposed estimator is developed based on Turing's formula, which recovers distributional characteristics on the subset of the alphabet not covered by a size-n sample. The fundamental switch in perspective brings about substantial gain in estimation accuracy for every distribution with finite entropy. In general, a uniform variance upper bound is established for the entire class of distributions with finite entropy that decays at a rate of O(ln(n)/n) compared to O([ln(n)]2/n) for the plug-in. In a wide range of subclasses, the variance of the proposed estimator converges at a rate of O(1/n), and this rate of convergence carries over to the convergence rates in mean squared errors in many subclasses. Specifically, for any finite alphabet, the proposed estimator has a bias decaying exponentially in n. Several new bias-adjusted estimators are also discussed.

A Simple Recursive Shannon Code

The Shannon code is a very simple suboptimal scheme for computing prefix-free codelengths for a given memoryless source. A recursive version of the well-known Shannon code (RSh) is investigated in this paper. It has a redundancy which never exceeds that of the Shannon code. Unlike the Huffman code, the RSh code and its variations can be directly applied to sources with an infinitely countable alphabet. The average redundancy, taken over the set of all n-tuple distributions, is considered as a criterion for code comparisons. For n>;40, the average redundancy is approximately 0.14 bits for the RSh code, compared to approximately 0.5 and 0.03 bits for the Shannon and Huffman codes, respectively. For large n, a constrained version of the RSh code, called CRSh, has an average redundancy of only 0.07 bits for unsorted symbol probabilities. If the symbol probabilities are sorted in descending order, then a very simple modification of the RSh algorithm results in a code which is near-optimal from the average redundancy point of view.

Adaptive compression against a countable alphabet

This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring (AC) code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy by Haussler and Opper (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate.

Toward the asymptotic count of bi-modular hidden patterns under probabilistic dynamical sources: a case study

Consider a countable alphabet $\mathcal{A}$. A multi-modular hidden pattern is an $r$-tuple $(w_1,\ldots , w_r)$, where each $w_i$ is a word over $\mathcal{A}$ called a module. The hidden pattern is said to occur in a text $t$ when the later admits the decomposition $t = v_0 w_1v_1 \cdots v_{r−1}w_r v_r$, for arbitrary words $v_i$ over $\mathcal{A}$. Flajolet, Szpankowski and Vallée (2006) proved via the method of moments that the number of matches (or occurrences) with a multi-modular hidden pattern in a random text $X_1\cdots X_n$ is asymptotically Normal, when $(X_n)_{n\geq1}$ are independent and identically distributed $\mathcal{A}$-valued random variables. Bourdon and Vallée (2002) had conjectured however that asymptotic Normality holds more generally when $(X_n)_{n\geq1}$ is produced by an expansive dynamical source. Whereas memoryless and Markovian sequences are instances of dynamical sources with finite memory length, general dynamical sources may be non-Markovian i.e. convey an infinite memory length. The technical difficulty to count hidden patterns under sources with memory is the context-free nature of these patterns as well as the lack of logarithm-and exponential-type transformations to rewrite the product of non-commuting transfer operators. In this paper, we address a case study in which we have successfully overpassed the aforementioned difficulties and which may illuminate how to address more general cases via auto-correlation operators. Our main result shows that the number of matches with a bi-modular pattern $(w_1, w_2)$ normalized by the number of matches with the pattern $w_1$, where $w_1$ and $w_2$ are different alphabet characters, is indeed asymptotically Normal when $(X_n)_{n\geq1}$ is produced by a holomorphic probabilistic dynamical source.

Multifractal Analysis for the Teichmüller Flow

We present a multifractal description for Teichmuller flows. A key ingredient to do this is the Rauzy–Veech–Zorich reduction theory, which allows to treat the problem in the setting of suspension flows over subshifts. To perform the multifractal analysis we implement a thermodynamic formalism for suspension flows over countable alphabet subshifts a bit different from that developed by Barreira and Iommi.

Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $\alpha$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundancy

Random graph directed Markov systems

We introduce and explore random conformal graph directed Mar-kovsystems governed by measure-preserving ergodic dynamicalsystems. We first develop the symbolic thermodynamicformalism for random finitely primitive subshifts of finite typewith a countable alphabet (by establishing tightness ina narrow topology). We then construct fibrewise conformal andinvariant measures along with fibrewise topological pressure.This enables us to define the expected topological pressure$\mathcal EP(t)$ and to prove a variant of Bowen's formulawhich identifies the Hausdorff dimension of almost everylimit set fiber with $\inf\{t:\mathcal EP(t)\leq0\}$, and is theunique zero of the expected pressure if the alphabet is finite orthe system is regular. We introduce the class of essentiallyrandom systems and we show that in the realm of systems withfinite alphabet their limit set fibers are never homeomorphic in abi-Lipschitz fashion to the limit sets of deterministic systems;they thus make up a drastically new world. We also provide alarge variety of examples, with exact computations of Hausdorffdimensions, and we study in detail the small random perturbationsof an arbitrary elliptic function.

On Information Divergence Measures and a Unified Typicality

Strong typicality, which is more powerful for theorem proving than weak typicality, can be applied to finite alphabets only, while weak typicality can be applied to countable alphabets. In this paper, the relation between typicality and information divergence measures is discussed. The new definition of information divergence measure in this paper leads to the definition of a unified typicality for finite or countably infinite alphabets which is stronger than both weak typicality and strong typicality. Unified typicality retains the asymptotic equipartition property and the structural properties of strong typicality, and it can potentially be used to generalize those theorems which are previously established by strong typicality to countable alphabets. The applications in rate-distortion theory and multisource network coding problems are discussed.

Weak KAM methods and ergodic optimal problems for countable Markov shifts

Let σ: Σ → Σ be the left shift acting on Σ, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of σ-invariant Borel probabilities that maximize the integral of a given locally Holder continuous potential A: Σ → ℝ. Under certain conditions, we are able to show not only that A-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).

A large deviations bound for the Teichmüller flow on the moduli space of abelian differentials

AbstractLarge deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base.Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems.Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow.Geom. Dedicata119(2006), 121–140].

The Erdős-Vershik problem for the golden ratio

Properties of the Erdős measure and the invariant Erdős measure for the golden ratio and all values of the Bernoulli parameter are studied. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdős measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdős measure is proposed. It is shown that, for certain values of the Bernoulli parameter, this algorithm gives the Hausdorff dimension of an Erdős measure to 15 decimal places.

Multifractal Decomposition of the Entropies Spectrum for Countable Subshifts in Finer Levels Sets

In a previous article we considered a finer decomposition of the spectrum of local entropies. In that case the dynamics are taken on compact spaces. Here we continue with this analysis but now for systems with noncompact phase space. Specifically, we consider here subshifts with countable alphabet.

On a symbolic dynamical zeta function and applications

A symbolic dynamical zeta function for subshifts over a countable alphabet is analyzed. We present examples on interval maps and suspensions.

Drunken man infinite words complexity

In this article, we study the complexity of drunken man infinite words. We show that these infinite words, generated by a deterministic and complete countable automaton, or equivalently generated by a substitution over a countable alphabet of constant length, have complexity functions equivalent to n(log 2 n) 2 when n goes to infinity.