From the divergence between two measures to the shortest path between two observables

Abadi, Miguel; Lambert, Rodrigo

doi:10.1017/etds.2017.114

Cited by 6 publications

(6 citation statements)

References 24 publications

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“…sequences of different lengths, different distributions, more than two sequences, extreme value theory for sequence matching) can be found in [6,9,7,8,33,20,37]. We also refer the reader to [43,36,3] for related sequence matching problems.…”

Section: Introductionmentioning

confidence: 99%

On the shortest distance between orbits and the longest common substring problem

Barros

Liao

Rousseau

2019

Advances in Mathematics

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In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of [5] on sequence matching to α-mixing processes with exponential decay.

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Section: Introductionmentioning

confidence: 99%

On the shortest distance between orbits and the longest common substring problem

Barros

Liao

Rousseau

2019

Advances in Mathematics

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“…Thus, taking a subsequence {n κ } κ = ⌈e κ 2 ⌉ as in the proof of (4), we can use Borel-Cantelli Lemma to obtain (5). Finally, if the Rényi entropy exists, by (4) and (5) we conclude the proof of the theorem.…”

Section: Reaching Rényi Entropy Via String Matching Of Encoded Sequencesmentioning

confidence: 74%

“…Thus, taking a subsequence {n κ } κ = ⌈e κ 2 ⌉ as in the proof of (4), we can use Borel-Cantelli Lemma to obtain (5).…”

Section: Reaching Rényi Entropy Via String Matching Of Encoded Sequencesmentioning

confidence: 99%

“…On the Erdös-Rényi scenario, the similarity between two sources has also been widely investigated (we refer the reader to [18,19,30,32] and references therein). As a recent example, we recall the shortest path between two observables defined in [5]. In this work the authors proved an almost-sure linear increasing, a large deviation principle and a weak convergence for the shortest path function.…”

Section: Introductionmentioning

confidence: 99%

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Matching strings in encoded sequences

Coutinho¹,

Lambert²,

Rousseau³

2020

Bernoulli

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We investigate the length of the longest common substring for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the Rényi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.keywords: string matching, Rényi entropy, shortest distance, correlation dimension, random dynamical systems, coding MSC: 60F15, 60Axx, 60C05, 37A50, 37A25, 37Hxx, 37C45, 94A17, 68P30

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“…In particular, we apply these results to the multidimensional expanding maps defined by Saussol [46]. Moreover, we also prove an annealed version of (2) for the shortest distance between k orbits of a random dynamical system. These results extend and complement those in [11] and [17] where identity (2) (and its equivalent for random dynamical systems) was proved for two orbits (k = 2).…”

Section: Introductionmentioning

confidence: 74%

Shortest distance between multiple orbits and generalized fractal dimensions

Barros¹,

Rousseau²

2019

Preprint

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We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy. Applications to Markov chains, Gibbs states and the stochastic scrabble are given.

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From the divergence between two measures to the shortest path between two observables

Abstract: We consider two independent and stationary measures over χ N , where χ is a finite or countable alphabet. For each pair of n-strings in the product space we define T(2) n Running head: The shortest path between two strings.Subject class: 37xx, 41A25, 60Axx , 60C05, 60Fxx.

Cited by 6 publications

References 24 publications

On the shortest distance between orbits and the longest common substring problem

On the shortest distance between orbits and the longest common substring problem

Matching strings in encoded sequences

Shortest distance between multiple orbits and generalized fractal dimensions

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