B. Vallee, Dynamical sources in information theory: fundamental intervals and word pre xes, Algorithmica 29 (2001), 262-306.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....TO iDYNAMICAL SOURCES IN INFORMATION THEORY: FUNDAMENTALS INTERVALS AND WORD PREFIXESj BY B. VALL#E FR#DERIC CHAZAL , V#RONIQUE MAUME DESCHAMPS , AND BRIGITTE VALL#E In [V], the third author studies statistical properties of words generated by dynamical sources by using generalised Ruelle operators. This erratum aims to show that a supplementary condition should be added in the denition 2.1 of dynamical sources. This supplementary condition (called (d4) below) is ....

....= h n (v) if and only if u = v 2ik log n ; k 2 Z shows that there does not exist any complex neighbourhood of I where all the inverse branches h are injective. 4. Existence of a dominant eigenvalue. As announced in the introduction, Property (d4) is also useful for proving Proposition 2 of [V]. In [V] this proposition claims the existence of a dominant eigenvalue for operators G s relative to real s. The main argument uses a result of Krasnoleskii and deals with the cone of real non negative analytic functions on V Theta V . Unfortunately, this cone does not satisfy the u 0 ....

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B. Vall#e, Dynamical sources in information theory: fundamental intervals and word prexes Algorithmica, 29, 262-306, (2001). *Universit# de Bourgogne B.P. 47870 21078 Dijon Cedex FRANCE E-mail address: fchazal@u-bourgogne.fr E-mail address: vmaume@u-bourgogne.fr ** GREYC, UMR CNRS 6072, Universit# de Caen, 14032 Caen, FRANCE E-mail address: Brigitte.Vallee@info.unicaen.fr

....under a probabilistic model on which the keys are i.i.d. random variables with density f on [0; 1] However, data on which tries are built often arise from real sources that may involve intricate dependencies between symbols. Here, we adopt the model of dynamical sources introduced by Vall ee in [25]. This model consists in a mechanism that associates a word M(x) to a real x of [0; 1] and an initial density f on [0; 1] The mechanism can be viewed as a limiting process of consecutive re nements of Markov chains that take into account a higher level of dependency on the symbols at each step. ....

....Mellin transform that is heavily used, the analysis involves the so called Dirichlet series of fundamental measures P w p s w . The main tool is then a generalization of Ruelle transfer operator that is used as a generating operator of fundamental probabilities. In previous papers, Vall ee [25], Cl ement [2] and Cl ement, Flajolet, Vall ee [3] have introduced successive generalizations of the Ruelle operator, mainly based on a secant (and multi secant) construction, that act on functions of two (or more) variables. Such operators depend on a complex parameter s and suitably generate ....

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Vall ee, B., Dynamical Sources in Information Theory: Fundamental Intervals and Word Prexes, Algorithmica (2001), vol 29 (1/2) pp 262-306 20

....the entropy function H ; the height estimate first appeared in [22] a paper already motivated by polynomial factorization) and it involves the coincidence probability 2 2 . Nowadays, these results are best understood in the context of Vall ee s general theory of dynamical sources; see [11, 58]. As a consequence of these estimates, splitting trees tend to be fairly well balanced so that the cost of an EDF phase is expected to be close to that of a perfect splitting. The lemma below provides an explicit expression for the costs induced by the computational model at hand. Lemma 5.1. The ....

Vall ee, B. Dynamical sources in information theory: fundamental intervals and word prefixes, Algorithmica 29 (2001), 262--306.

.... that the statement is fairly robust: it extends to a somewhat larger class of patterns with gaps; more importantly perhaps, concentration of distribution is shown in [9] to hold for a wide class of sources encompassing memoryless and Markov sources the dynamical sources in the sense of Vall ee [10, 39]. 3.3. Generalized autocorrelations and variances. In this subsection, we reexamine the variance coecient, for which formul have been provided earlier; see (14) of Theorem 1. As we now explain, the variance coecient turns out to be computable in a time that is polynomial in the size of the ....

B. Vallee, Dynamical Sources in Information Theory: Fundamental Intervals and Word Pre- xes, Algorithmica, 29, 262-306, 2001.

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B. Vallee, Dynamical sources in information theory: fundamental intervals and word pre xes, Algorithmica 29 (2001), 262-306.

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B. Vallee, Dynamical Sources in Information Theory : Fundamental intervals and Word Pre xes, Algorithmica, 29, 262-306, 2001.

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Vall#e, B. Dynamical sources in information theory: Fundamental intervals and words prexes. Technical report of the University of Caen, Les cahiers du GREYC, 2 (1999). Submitted.

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