Le théorème limite central pour les suites de R. C. Baker

Fukuyama, Katusi; Petit, Bernard

doi:10.1017/s0143385701001237

Cited by 15 publications

(20 citation statements)

References 4 publications

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“…To conclude this introduction, let us mention that in a previous work [5] we extended to ergodic sums of multidimensional actions by endomorphisms the CLT proved by T. Fukuyama and B. Petit [14] for coprime integers acting on the circle. After completing it, we were informed of the results of M. Levin [26] showing the CLT for ergodic sums over rectangles for actions by endomorphisms on tori.…”

Section: Introductionmentioning

confidence: 99%

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Cohen

Conze

2015

J Theor Probab

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Abstract. Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system of generators (In particular, given a random walk on commuting matrices in SL(ρ, Z) or in M * (ρ, Z) acting on the torus T ρ , ρ ≥ 1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on T ρ after normalization?In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g. a torus or a connected extension of a torus), S a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums.

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Section: Introductionmentioning

confidence: 99%

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Cohen

Conze

2015

J Theor Probab

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“…. , n N ) was computed by Fukuyama and Petit [9]; in Section 3 we will determine the class of limits…”

Section: Hence In This Casementioning

confidence: 99%

“…As mentioned, for the original, unpermuted sequence (n k ), the value of γ = γ f in (1.8) was computed in [9]. Given an f satisfying condition (1.3), let Γ f denote the set of limiting variances in (1.8) belonging to all permutations σ.…”

Section: Hence In This Casementioning

confidence: 99%

On permutations of Hardy-Littlewood-Pólya sequences

Aistleitner

Berkes

Tichy

2011

Trans. Amer. Math. Soc.

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Abstract. Let H = (q 1 , . . . , q r ) be a finite set of coprime integers and let n 1 , n 2 , . . . denote the multiplicative semigroup generated by H and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence {n k x} are similar to those of independent, identically distributed random variables; here {·} denotes fractional part. In this paper we prove that under mild assumptions on the periodic function f , the sequence f (n k x) obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences f (m k x) even if (m k ) has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that {n k x} satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums S N (σ) = N k=1 f (n σ(k) x).

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“…For the CLT for trigonometric series with small gaps see Berkes [4] and Bobkov and Götze [6] introducing a completely new method in gap theory. For recent asymptotic results for f (n k x) for subexponential (n k ) k≥1 see e.g., Philipp [20], Fukuyama and Petit [11] and Aistleitner and Berkes [1].…”

Section: Where C(d) Is a Constant Depending Only On Dmentioning

confidence: 99%

On the central limit theorem for f (n k x)

Aistleitner

Berkes

2008

Probab. Theory Relat. Fields

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By a classical observation in analysis, lacunary subsequences of the trigonometric system behave like independent random variables: they satisfy the central limit theorem, the law of the iterated logarithm and several related probability limit theorems. For subsequences of the system ( f (nx)) n≥1 with 2π -periodic f ∈ L 2 this phenomenon is generally not valid and the asymptotic behavior of ( f (n k x)) k≥1 is determined by a complicated interplay between the analytic properties of f (e.g., the behavior of its Fourier coefficients) and the number theoretic properties of n k . By the classical theory, the central limit theorem holds for f (n k x) if n k = 2 k , or if n k+1 /n k → α with a transcendental α, but it fails e.g., for n k = 2 k − 1. The purpose of our paper is to give a necessary and sufficient condition for f (n k x) to satisfy the central limit theorem. We will also study the critical CLT behavior of f (n k x), i.e., the question what happens when the arithmetic condition of the central limit theorem is weakened "infinitesimally".

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Le théorème limite central pour les suites de R. C. Baker

Abstract: Let D = (ω n ) n≥0 be the multiplicative semi-group generated by the coprime integers q 1 , . . . , q τ arranged in increasing order. If f is a real-valued 1-periodic function, we consider the sums S n f (t) = 0≤k

Cited by 15 publications

References 4 publications

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

On permutations of Hardy-Littlewood-Pólya sequences

On the central limit theorem for f (n k x)

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