Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

Sera, Toru; Yano, Kouji

doi:10.1090/tran/7755

Cited by 11 publications

(18 citation statements)

References 31 publications

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“…[TZ, SY] and references therein. A strong recent result is that of Sera and Yano [SY,Thm. 2.7] about the joint distribution of the frequencies of visits to many infinite-measure sets.…”

Section: Counterexamples and Discussionmentioning

confidence: 98%

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Pointwise convergence of Birkhoff averages for global observables

Lenci

Munday

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system, the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper, we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of , which we generically call global observables. Our main theorem applies to general systems but requires a hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as we show with an example. Finally, by means of counterexamples and numerical simulations, we discuss the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems.

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“…[TZ, SY] and references therein. A strong recent result is that of Sera and Yano [SY,Thm. 2.7] about the joint distribution of the frequencies of visits to many infinite-measure sets.…”

Section: Counterexamples and Discussionmentioning

confidence: 98%

“…Nevertheless, the statistical properties of A n f cannot be derived from the main theorem of [SY], and not only because here we have infinitely many rays. The most important difference is that the assumption on the asymptotic entrance densities [SY,Ass. 2.3] is not satisfied.…”

Section: Counterexamples and Discussionmentioning

confidence: 99%

Pointwise convergence of Birkhoff averages for global observables

Lenci

Munday

2018

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…We obtain the scaling limit of the time evolution in the sense of strong distributional convergence. Our limit theorem is a functional and joint-distributional extension of Darling-Kac type limit theorems [2,3,51,59,29,37], of Lamperti type generalized arcsine laws for occupation times [50,51,59,45], and of Dynkin and Lamperti type generalized arcsine laws for waiting times [49,51,59,29], at the same time.…”

Section: Introductionmentioning

confidence: 75%

“…Aaronson [3] and Owada-Samorodnitsky [37] obtained a functional extension of Darling-Kac type. In the previous study [45], the author and Kouji Yano obtained a multiray generalization of Lamperti type. They focused on interval maps with more than two indifferent fixed points, and studied the joint-law of occupations near each of these points.…”

Section: Introductionmentioning

confidence: 99%

Functional limit theorem for occupation time processes of intermittent maps

Sera¹

2018

Preprint

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We establish a functional limit theorem for the joint-law of occupations near and away from indifferent fixed points of interval maps, and of waits for the occupations away from these points, in the sense of strong distributional convergence. It is a functional and joint-distributional extension of Darling-Kac type limit theorem, of Lamperti type generalized arcsine laws for occupation times, and of Dynkin and Lamperti type generalized arcsine laws for waiting times, at the same time.

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“…In particular, distribution of time averages of L 1 (µ) function, i.e., a function integrable with respect to invariant measure µ, converge to the Mittag-Leffler distribution [22,23]. These distributional behaviors of time averages are characteristics of infinite ergodic theory, which includes the Mittag-Leffler distribution, the generalized arcsine distribution and another distribution [24][25][26][27][28][29][30].…”

mentioning

confidence: 99%

Aging arcsine law in Brownian motion and its generalization

Akimoto,

Sera,

Yamato

et al. 2020

Preprint

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Classical arcsine law states that fraction of occupation time on the positive or the negative side in Brownian motion does not converge to a constant but converges in distribution to the arcsine distribution. Here, we consider how a preparation of the system affects the arcsine law, i.e., aging of the arcsine law. We derive aging distributional theorem for occupation time statistics in Brownian motion, where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution. Furthermore, we show that this result can be generalized as aging distributional limit theorem in renewal processes.

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Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

Cited by 11 publications

References 31 publications

Pointwise convergence of Birkhoff averages for global observables

Pointwise convergence of Birkhoff averages for global observables

Functional limit theorem for occupation time processes of intermittent maps

Aging arcsine law in Brownian motion and its generalization

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