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Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System

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Abstract

Limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied. It is known that intermittent phenomena, such as the Rayleigh-Benard convection and Belousov-Zhabotinsky reaction, are described by infinite measure dynamical systems. We show that the time average of the observation function which is not the L 1(m) function, whose average with respect to the invariant measure m is finite, converges to the generalized arcsine distribution. This result leads to the novel view that the correlation function is intrinsically random and does not decay. Moreover, it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean.

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Authors and Affiliations

  1. Department of Applied Physics, Advanced School of Science and Engineering, Waseda University, Okubo 3-4-1, Shinjuku-ku, Tokyo, 169-8555, Japan

    Takuma Akimoto

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  1. Takuma Akimoto
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Correspondence to Takuma Akimoto.

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Akimoto, T. Generalized Arcsine Law and Stable Law in an Infinite Measure Dynamical System. J Stat Phys 132, 171–186 (2008). https://doi.org/10.1007/s10955-008-9544-9

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  • DOI: https://doi.org/10.1007/s10955-008-9544-9

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