A Nonconventional Local Limit Theorem

Hafouta, Yeor; Kifer, Yuri

doi:10.1007/s10959-015-0625-9

Cited by 21 publications

(16 citation statements)

References 29 publications

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“…Our proof there was based on a certain reduction to a problem of bounding expectations of norms of iterates of random Fourier operators (the proof was in the spirit of the argument in [26]). This is exactly the situation of annealed limit theorems and so, similar to [13], the argument will yield the desired results.…”

Section: Y Hafoutamentioning

confidence: 76%

Limit theorems for some skew products with mixing base maps

Hafouta¹

2019

Ergod. Th. Dynam. Sys.

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We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps T (ω, x) = (θω, Tωx) together with a T -invariant measure, whose base map θ satisfies certain topological and mixing conditions and the maps Tω on the fibers are certain non-singular distance expanding maps. Our results hold true when θ is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle-Perron-Frobenius theorem from [13] applied with appropriate random transfer operators generated by Tω, together with certain regularity assumptions (as functions of ω) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition will also be obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in [1] does not seem to be applicable, and the dual of the Koopman operator of T (with respect to the invariant measure) does not seem to have a spectral gap.

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Section: Y Hafoutamentioning

confidence: 76%

Limit theorems for some skew products with mixing base maps

Hafouta¹

2019

Ergod. Th. Dynam. Sys.

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“…Our results below can be obtained assuming that X(n) is measurable with respect to F n,n without special assumptions on the function F beyond measurability similarly to [10]. Nevertheless, we prefer here the setup from [14] which allows applications to dynamical systems.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…On the other hand, the limiting behavior of the variance ξ N in (1.2) was not studied in [14] in spite of the fact that a meaningful central limit theorem requires the limit of Varξ N as N → ∞ to be positive. Some partial results in this direction were obtained in [12] and [10]. Ensuring positivity of the limiting variance and obtaining appropriate lower bounds for it is especially important in Berry-Esseen type estimates and we provide here a rather complete answer concerning this question.…”

Section: Introductionmentioning

confidence: 83%

Berry–Esseen type estimates for nonconventional sums

Hafouta

Kifer

2016

Stochastic Processes and their Applications

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Abstract. We obtain Berry-Esseen type estimates for "nonconventional" expressions of the formwhere X(n) is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties,F = F d(µ × ... × µ), µ is the distribution of X(0) and q i (n) = in for 1 ≤ i ≤ k while for i > k they are positive functions taking integer values on integers with some growth conditions which are satisfies, for instance, when they are polynomials of increasing degrees. Our setup is similar to [14] where a nonconventional functional central limit theorem was obtained and the present paper provides estimates for the convergence speed. As a part of the study we provide answers for the crucial question on positivity of the limiting variance lim N→∞ Var(ξ N ) which was not studied in [14]. Extensions to the continuous time case will be discussed as well. As in [14] our results are applicable to stationary processes generated by some classes of sufficiently well mixing Markov chains and dynamical systems.

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“…Its proof is based on the inversion formula for Fourier transform, which is a traditional argument for this type of behavior. Its statement is practically obtained by arguments in Section 4 in Hafouta and Kifer (2016).…”

Section: 2mentioning

confidence: 99%

“…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). Hafouta and Kifer (2016) proved a local limit theorem for nonconventional sums for a class of stationary Markov chains.…”

Section: Introductionmentioning

confidence: 99%