Stable functional CLT for deterministic systems

Zemer Kosloff; Dalibor Volný

doi:10.1017/etds.2025.17

Stable functional CLT for deterministic systems

Zemer Kosloff^*, Dalibor Volný

^*Corresponding author for this work

Einstein Institute of Mathematics

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

Abstract

We show that α-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 0 < α < 1 and every α-stable Lévy motion W, there exists a function f whose partial sum process converges in distribution to W; for 1 ≤ α < 2 and every symmetric α-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to W; for 1 < α < 2 and every −1 ≤ β ≤ 1 there exists a function f whose associated time series is in the classical domain of attraction of an S_α(ln(2), β, 0) random variable.

Original language	English
Journal	Ergodic Theory and Dynamical Systems
DOIs	https://doi.org/10.1017/etds.2025.17
State	Accepted/In press - 2025

Bibliographical note

Publisher Copyright:
© The Author(s), 2025. Published by Cambridge University Press.

Keywords

dynamical systems
limit theorems
stable processes
weak convergence

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10.1017/etds.2025.17

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/49260DA376DECDDFCE75F4D4640A76DD/S0143385725000173a.pdf/div-class-title-stable-functional-clt-for-deterministic-systems-div.pdfLicense: CC BY

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abstract = "We show that α-stable L{\'e}vy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 0 < α < 1 and every α-stable L{\'e}vy motion W, there exists a function f whose partial sum process converges in distribution to W; for 1 ≤ α < 2 and every symmetric α-stable L{\'e}vy motion, there exists a function f whose partial sum process converges in distribution to W; for 1 < α < 2 and every −1 ≤ β ≤ 1 there exists a function f whose associated time series is in the classical domain of attraction of an Sα(ln(2), β, 0) random variable.",

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doi = "10.1017/etds.2025.17",

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AU - Kosloff, Zemer

AU - Volný, Dalibor

N1 - Publisher Copyright: © The Author(s), 2025. Published by Cambridge University Press.

PY - 2025

Y1 - 2025

N2 - We show that α-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 0 < α < 1 and every α-stable Lévy motion W, there exists a function f whose partial sum process converges in distribution to W; for 1 ≤ α < 2 and every symmetric α-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to W; for 1 < α < 2 and every −1 ≤ β ≤ 1 there exists a function f whose associated time series is in the classical domain of attraction of an Sα(ln(2), β, 0) random variable.

AB - We show that α-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 0 < α < 1 and every α-stable Lévy motion W, there exists a function f whose partial sum process converges in distribution to W; for 1 ≤ α < 2 and every symmetric α-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to W; for 1 < α < 2 and every −1 ≤ β ≤ 1 there exists a function f whose associated time series is in the classical domain of attraction of an Sα(ln(2), β, 0) random variable.

KW - dynamical systems

KW - limit theorems

KW - stable processes

KW - weak convergence

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SN - 0143-3857

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

ER -

Stable functional CLT for deterministic systems

Abstract

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Keywords

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