Large-deviation probability and the local dimension of sets

Barbe, Philippe; Broniatowski, Michel

doi:10.1007/bf02674081

Cited by 2 publications

(9 citation statements)

References 12 publications

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“…Let µ := lim t→∞ µ ′ 1 (t) and µ := lim t→0 µ ′ 1 (t). Then according to [2] the following holds: Lemma 16. Under the above notation and hypotheses, the equation Ψ ′ n (t) = 0 has a unique solution t n in (0, t 0 ) for α in (EU + µ, ∞) where t 0 := sup{t : φ U (t) < ∞}.…”

Section: 1mentioning

confidence: 99%

“…where t n is a solution of Ψ ′ n (t) = 0 in the range (0, t 0 ). It can be proved that (41) holds, for example, when t → log M (t)/t is a regularly varying function at infinity with index ρ ∈ (0, 1), that is, log M (t)/t ∈ R ρ (∞); see [2], Lemma 2.2.…”

Section: 1mentioning

confidence: 99%

“…Theorem 2.1 in [2] provides a general result to be inserted in (40); we take the occasion to correct a misprint in this result.…”

Section: 1mentioning

confidence: 99%

“…but through the more appropriate quantity for all t > 0 such that φ U (t) is finite. We borrow from [2] the following results. Define µ n (t) := (1/n) log M n (t) which is for all n ≥ 1 a decreasing function of t on (0, ∞), and which is negative for large n. Also µ ′ n (t) = µ ′ 1 (nt) and µ ′ 1 are nondecreasing on (0, ∞).…”

mentioning

confidence: 99%

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Long runs under a conditional limit distribution

Broniatowski¹,

Caron²

2014

Ann. Appl. Probab.

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This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a function of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with a nonempty interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. An application of the approximation scheme to the evaluation of rare event probabilities through importance sampling is provided. When the conditioning event is in the range of the central limit theorem, it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao-Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.Comment: Published in at http://dx.doi.org/10.1214/13-AAP975 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1010.361

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Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

“…Theorem 2.1 in [2] provides a general result to be inserted in (40); we take the occasion to correct a misprint in this result.…”

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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Long runs under a conditional limit distribution

Broniatowski¹,

Caron²

2014

Ann. Appl. Probab.

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“…Andriani and Baldi [1] recover this equivalent and focus on the geometric meaning of the terms involved. We mention that Barbe and Broniatowski [3] settle the question for sums of i.i.d. random variables and arbitrary Borel sets B. Kontoyiannis and Meyn [14] prove an equivalent of the pre-exponent term for Markov-additive processes in dimension d = 1, when the underlying Markov chain lives on a general state space and is geometrically ergodic.…”

Section: Exact Asymptotic Equivalentsmentioning

confidence: 99%

Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

Pudlo

2010

ESAIM: PS

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Abstract.To establish lists of words with unexpected frequencies in long sequences, for instance in a molecular biology context, one needs to quantify the exceptionality of families of word frequencies in random sequences. To this aim, we study large deviation probabilities of multidimensional word counts for Markov and hidden Markov models. More specifically, we compute local Edgeworth expansions of arbitrary degrees for multivariate partial sums of lattice valued functionals of finite Markov chains. This yields sharp approximations of the associated large deviation probabilities. We also provide detailed simulations. These exhibit in particular previously unreported periodic oscillations, for which we provide theoretical explanations.Mathematics Subject Classification. 60J10, 60F10, 60J55, 92D20, 60F05.

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Large-deviation probability and the local dimension of sets

Cited by 2 publications

References 12 publications

Long runs under a conditional limit distribution

Long runs under a conditional limit distribution

Large deviations and full Edgeworth expansions for finite Markov chains with applications to the analysis of genomic sequences

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