A Local Limit Theorem for Continued Fractions

Szewczak, Zbigniew S.

doi:10.1142/s0219493710002954

Cited by 6 publications

(6 citation statements)

References 29 publications

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“…As far as we know this corollary is new, though for Gibbs-Markov processes the result is provided in Aaronson and Denker (2001a) and for continued fraction processes can be found in Szewczak (2010).…”

Section: Resultsmentioning

confidence: 95%

“…Concerning dependent random variables we should mention early works on Markov chains by Kolmogorov (1962). In the lattice case, for countable state Markov chains with finite second moments, the local limit theorem is discussed in Siraždinov (1955) and Séva (1995) while the case of infinite variance is analyzed in Aaronson and Denker (2001a) and Szewczak (2008Szewczak ( , 2010. Also in the stationary case we mention the local limit theorems for Markov chains in the papers by Hervé and Pène (2010) and Ferré et al (2012).…”

Section: Introductionmentioning

confidence: 99%

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On the local limit theorems for psi-mixing Markov chains

Merlevède¹,

Peligrad²,

Peligrad³

2021

ALEA

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Section: Resultsmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

On the local limit theorems for psi-mixing Markov chains

Merlevède¹,

Peligrad²,

Peligrad³

2021

ALEA

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“…As far as we know this corollary is new, though for Gibbs-Markov processes the result is contained in Aaronson and Denker (2001a) and for continued fraction processes can be found in Szewczak (2010).…”

Section: Local Central Limit Theoremmentioning

confidence: 94%

“…Dolgopyat (2016) treated the vector valued sequences of independent random variables. In the lattice case, for countable state Markov chains with finite second moments, the local limit theorem is discussed in Nagaev (1963) and Séva (1995) while the case of infinite variance is analyzed in Aaronson and Denker (2001a,b) and Szewczak (2010).…”

Section: Introductionmentioning

confidence: 99%

On the local limit theorems for psi-mixing Markov chains

Merlevède¹,

Peligrad²,

Peligrad³

2020

Preprint

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“…The proof of Theorem 1.95 (see [219] for details) does not make use of the spectral perturbation method (cf. [173], [204], [97]).…”

Section: The Ergodic Casementioning

confidence: 99%

Classical and Almost Sure Local Limit Theorems

Szewczak¹,

Weber²

2022

Preprint

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In this monograph, we present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem, which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly concern sufficient conditions for the validity of the local limit theorem and its interesting variant forms: strong local limit theorem, strong local limit theorem with convergence in variation. Quite importantly are necessary conditions, and the results obtained are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary condition, and, almost isolated among the flow of results, Mukhin's necessary and sufficient condition. Extremely useful and instructive are the counter-examples due to Azlarov and Gamkrelidze, as well as necessary and sufficient conditions obtained for a class of random variables, such as Mitalauskas' characterization of the local limit theorem in the strong form for random variables having stable limit distributions. The method of characteristic functions and the Bernoulli part extraction method, are presented and compared. A second part of the survey is devoted to the more recent study of the almost sure local limit theorem, instilled by Denker and Koch. The inherent second order study, which has its own interest, is much more difficult than for establishing the almost sure central limit theorem. The almost sure local limit theorems established already cover the i.i.d. case, the stable case, Markov chains, the model of the Dickman function, and the independent case, with almost sure convergence of related series.Our aim in writing this monograph was first to share the passion resulting from the study for this very special limit theorem, second to bring to knowledge many interesting results obtained by the Lithuanian and Russian Schools of Probability during the sixties and after, and which are essentially written in Russian, and moreover often published in Journals of difficult access. In doing so, our feeling was to somehow help with this whole coherent body of results and methods, researchers in the study of the local limit theorem, at least it is our hope.

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A Local Limit Theorem for Continued Fractions

Abstract: It is shown that functionals of digits in continued fraction expansion satisfy either the DeMoivre-Gnedenko or the Shepp-Stone limit theorems if and only if their marginals are in the domain of attraction of the normal law.

Cited by 6 publications

References 29 publications

On the local limit theorems for psi-mixing Markov chains

On the local limit theorems for psi-mixing Markov chains

On the local limit theorems for psi-mixing Markov chains

Classical and Almost Sure Local Limit Theorems

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