Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

Kesseböhmer, Marc; Schindler, Tanja I.

doi:10.1088/1361-6544/ab9585

Cited by 7 publications

(8 citation statements)

References 31 publications

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“…Remark 3.1. Indeed by [KS3] the stronger result of convergence in mean follows for c). It is not proven that for the situation in c) convergence in probability can not hold for a lightly trimmed sum, i.e.…”

Section: Main Results -Distributional Mattersmentioning

confidence: 93%

Prime numbers in typical Continued Fraction Expansions

Schindler¹,

Zweimüller²

2022

Preprint

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Section: Main Results -Distributional Mattersmentioning

confidence: 93%

Prime numbers in typical Continued Fraction Expansions

Schindler¹,

Zweimüller²

2022

Preprint

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“…Kesseböhmer and the second author of this paper proved strong laws of large numbers under intermediate trimming using a spectral gap property of the transfer operator, see [KS19b] and [KS20a] for an application of these results to subshifts of finite type and [HM87, H93,KS19a] for these and further reaching results in the independent case. See further [KS20b] for a convergence in mean result concerning the same intermediately trimmed sums.…”

Section: Introductionmentioning

confidence: 93%

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Bonanno¹,

Schindler²

2021

Preprint

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We consider a conservative ergodic measure-preserving transformation T of a σ-finite measure space (X, B, µ) with µ(X) = ∞. Given an observable f : X → R we study the almost sure asymptotic behaviour of the Birkhoff sums SN f (x) := N j=1 (f •T j−1 )(x). In infinite ergodic theory it is well known that the asymptotic behaviour of SN f (x) strongly depends on the point x ∈ X, and if f ∈ L 1 (X, µ), then there exists no real valued sequence (b(N )) such that limN→∞ SN f (x)/b(N ) = 1 almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence (α(N )) and m : X × N → N such that for f ∈ L 1 (X, µ) we have limN→∞ S N+m(x,N) f (x)/α(N ) = 1 for µ-a.e. x ∈ X. Moreover if f ∈ L 1 (X, µ) we give conditions on the induced observable such that there exists a sequence (G(N )) depending on f , for which limN→∞ SN f (x)/G(N ) = 1 holds for µ-a.e. x ∈ X.

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“…On the other hand, let's consider a global observable f : X → R ≥0 which is constant on the level sets (E n ) with f n := f | En . By rescaling we can assume that the limit in (25) exists and is equal to 1. In general it is not possible to conclude about the asymptotic behaviour of f E | An , however some sufficient conditions can be obtained by the following argument (see [29]).…”

Section: 2mentioning

confidence: 99%

“…Kesseböhmer and the second author of this paper proved strong laws of large numbers under intermediate trimming using a spectral gap property of the transfer operator, see [23] and [24] for an application of these results to subshifts of finite type and [17,16,22] for these and further reaching results in the independent case. See further [25] for a convergence in mean result concerning the same intermediately trimmed sums.…”

mentioning

confidence: 93%

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Bonanno

Schindler

2022

DCDS

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<p style='text-indent:20px;'>We consider a conservative ergodic measure-preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of a <inline-formula><tex-math id="M2">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-finite measure space <inline-formula><tex-math id="M3">\begin{document}$ (X, {\mathcal B},\mu) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ \mu(X) = \infty $\end{document}</tex-math></inline-formula>. Given an observable <inline-formula><tex-math id="M5">\begin{document}$ f:X\to \mathbb R $\end{document}</tex-math></inline-formula>, we study the almost sure asymptotic behaviour of the Birkhoff sums <inline-formula><tex-math id="M6">\begin{document}$ S_Nf(x) : = \sum_{j = 1}^N\, (f\circ T^{j-1})(x) $\end{document}</tex-math></inline-formula>. In infinite ergodic theory it is well known that the asymptotic behaviour of <inline-formula><tex-math id="M7">\begin{document}$ S_Nf(x) $\end{document}</tex-math></inline-formula> strongly depends on the point <inline-formula><tex-math id="M8">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>, and if <inline-formula><tex-math id="M9">\begin{document}$ f\in L^1(X,\mu) $\end{document}</tex-math></inline-formula>, then there exists no real valued sequence <inline-formula><tex-math id="M10">\begin{document}$ (b(N)) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M11">\begin{document}$ \lim_{N\to\infty} S_Nf(x)/b(N) = 1 $\end{document}</tex-math></inline-formula> almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence <inline-formula><tex-math id="M12">\begin{document}$ (\alpha(N)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ m\colon X\times \mathbb N\to \mathbb N $\end{document}</tex-math></inline-formula> such that for <inline-formula><tex-math id="M14">\begin{document}$ f\in L^1(X,\mu) $\end{document}</tex-math></inline-formula> we have <inline-formula><tex-math id="M15">\begin{document}$ \lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N) = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M17">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>. Instead in the case <inline-formula><tex-math id="M18">\begin{document}$ f\not\in L^1(X,\mu) $\end{document}</tex-math></inline-formula> we give conditions on the induced observable such that there exists a sequence <inline-formula><tex-math id="M19">\begin{document}$ (G(N)) $\end{document}</tex-math></inline-formula> depending on <inline-formula><tex-math id="M20">\begin{document}$ f $\end{document}</tex-math></inline-formula>, for which <inline-formula><tex-math id="M21">\begin{document}$ \lim_{N\to\infty} S_{N}f(x)/G(N) = 1 $\end{document}</tex-math></inline-formula> holds for <inline-formula><tex-math id="M22">\begin{document}$ \mu $\end{document}</tex-math></inline-formula>-a.e. <inline-formula><tex-math id="M23">\begin{document}$ x\in X $\end{document}</tex-math></inline-formula>.</p>

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Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails

Cited by 7 publications

References 31 publications

Prime numbers in typical Continued Fraction Expansions

Prime numbers in typical Continued Fraction Expansions

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

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