Local Limit Theorems for Stable Limit Distributions

Mitalauskas, A.

doi:10.1137/1107016

Cited by 4 publications

(3 citation statements)

References 2 publications

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“…For ρ = ρ c the sub-exponential tails of ν 1 1 can lead to diverging second moments, but in any case the local limit theorem for non-normal limit distributions (see e.g. [48]) provides a lower bound of order ν Λ φ Σ Λ = N ≥ 1/L which leads to the same conclusion. For ρ > ρ c this is a large deviation probability, and a simple lower bound is given by putting all excess mass in the first site,…”

Section: General Resultsmentioning

confidence: 81%

Condensation in Stochastic Particle Systems with Stationary Product Measures

2013

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We study stochastic particle systems with stationary product measures that exhibit a condensation transition due to particle interactions or spatial inhomogeneities. We review previous work on the stationary behaviour and put it in the context of the equivalence of ensembles, providing a general characterization of the condensation transition for homogeneous and inhomogeneous systems in the thermodynamic limit. This leads to strengthened results on weak convergence for subcritical systems, and establishes the equivalence of ensembles for spatially inhomogeneous systems under very general conditions, extending previous results which focused on attractive and finite systems. We use relative entropy techniques which provide simple proofs, making use of general versions of local limit theorems for independent random variables. This paper is dedicated to Herbert Spohn in honour of his 65th birthday.

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Section: General Resultsmentioning

confidence: 81%

Condensation in Stochastic Particle Systems with Stationary Product Measures

2013

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“…Remark 1.24. The terminology 'in the strong form', also 'in strong sense', is used by various authors, for instance in Gamkrelidze [73], [66], Mitalauskas [156], Mitalauskas-Stepanauskas [157], Mukhin [167], . .…”

Section: The Independent Casementioning

confidence: 99%

“…. It was also proved that under relatively general supplementary conditions in [186], [196], [156] notably, see also Theorems 1.37, 1.38, 1.40, 1.41, from the integral limit theorem and a.u.d. property for partial sums follows the local limit theorem.…”

Section: The Independent 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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Accuracy of approximation of the distribution functions of the sums of independent random variables by stable distributions

Kalinauskaitė

1974

Lith Math J

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Local Limit Theorems for Stable Limit Distributions

Cited by 4 publications

References 2 publications

Condensation in Stochastic Particle Systems with Stationary Product Measures

Condensation in Stochastic Particle Systems with Stationary Product Measures

Classical and Almost Sure Local Limit Theorems

Accuracy of approximation of the distribution functions of the sums of independent random variables by stable distributions

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