### Abstract

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of L^{1}(m) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-L^{1}(m) functions. Here, we provide another distributional behavior of time averages of non-L^{1}(m) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-L^{1}(m) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-L^{1}(m) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-L^{1}up>(m) function in the one-dimensional intermittent maps.

Original language | English |
---|---|

Pages (from-to) | 476-493 |

Number of pages | 18 |

Journal | Journal of Statistical Physics |

Volume | 158 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2015 Jan 1 |

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### Keywords

- Anomalous diffusion
- Distributional limit theorem
- Infinite ergodic theory

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

^{1}Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures.

*Journal of Statistical Physics*,

*158*(2), 476-493. https://doi.org/10.1007/s10955-014-1138-0

**Distributional Behavior of Time Averages of Non-L ^{1} Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures.** / Akimoto, Takuma; Shinkai, Soya; Aizawa, Yoji.

Research output: Contribution to journal › Article

^{1}Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures',

*Journal of Statistical Physics*, vol. 158, no. 2, pp. 476-493. https://doi.org/10.1007/s10955-014-1138-0

^{1}Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures. Journal of Statistical Physics. 2015 Jan 1;158(2):476-493. https://doi.org/10.1007/s10955-014-1138-0

}

TY - JOUR

T1 - Distributional Behavior of Time Averages of Non-L1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

AU - Akimoto, Takuma

AU - Shinkai, Soya

AU - Aizawa, Yoji

PY - 2015/1/1

Y1 - 2015/1/1

N2 - In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of L1(m) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-L1(m) functions. Here, we provide another distributional behavior of time averages of non-L1(m) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-L1(m) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-L1(m) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-L1up>(m) function in the one-dimensional intermittent maps.

AB - In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of L1(m) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-L1(m) functions. Here, we provide another distributional behavior of time averages of non-L1(m) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-L1(m) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-L1(m) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-L1up>(m) function in the one-dimensional intermittent maps.

KW - Anomalous diffusion

KW - Distributional limit theorem

KW - Infinite ergodic theory

UR - http://www.scopus.com/inward/record.url?scp=84921978308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84921978308&partnerID=8YFLogxK

U2 - 10.1007/s10955-014-1138-0

DO - 10.1007/s10955-014-1138-0

M3 - Article

AN - SCOPUS:84921978308

VL - 158

SP - 476

EP - 493

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -