A Nonconventional Local Limit Theorem

Yeor Hafouta; Yuri Kifer

doi:10.1007/s10959-015-0625-9

A Nonconventional Local Limit Theorem

Yeor Hafouta
, Yuri Kifer^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as N→ ∞ of probabilities having the form P{ S_N= k} where SN=∑n=1NF(ξn) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables { ξ_j}. Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form SN=∑n=1NF(ξn,ξ2n,…,ξℓn) which continues the recent line of research studying various limit theorems for such expressions.

Original language	English
Pages (from-to)	1524-1553
Number of pages	30
Journal	Journal of Theoretical Probability
Volume	29
Issue number	4
DOIs	https://doi.org/10.1007/s10959-015-0625-9
State	Published - 1 Dec 2016

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.

Keywords

Local limit theorem
Markov chain
Mixing
Nonconventional setup

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10.1007/s10959-015-0625-9

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N2 - Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as N→ ∞ of probabilities having the form P{ SN= k} where SN=∑n=1NF(ξn) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables { ξj}. Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form SN=∑n=1NF(ξn,ξ2n,…,ξℓn) which continues the recent line of research studying various limit theorems for such expressions.

AB - Local limit theorems have their origin in the classical De Moivre–Laplace theorem, and they study the asymptotic behavior as N→ ∞ of probabilities having the form P{ SN= k} where SN=∑n=1NF(ξn) is a sum of an integer-valued function F taken on i.i.d. or Markov-dependent sequence of random variables { ξj}. Corresponding results for lattice-valued and general functions F were obtained, as well. We extend here this type of results to nonconventional sums of the form SN=∑n=1NF(ξn,ξ2n,…,ξℓn) which continues the recent line of research studying various limit theorems for such expressions.

KW - Local limit theorem

KW - Markov chain

KW - Mixing

KW - Nonconventional setup

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A Nonconventional Local Limit Theorem

Abstract

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Keywords

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