2013/10/31 by Takuma Akimoto, Soya Shinkai, Yoji Aizawa · 22 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Class (philosophy) #Distribution (mathematics) #Ergodic theory #Fixed point #Integrable system #Invariant (physics) #Invariant measure #Limit (mathematics) #Mathematical Dynamics and Fractals #Stochastic process #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.DS #nlin.CD
paper · pdf · doi:10.1007/s10955-014-1138-0
published in Journal of Statistical Physics 158(2), 476-493 (Springer Science+Business Media) · 24 pages, 6 figures
arxiv created 2014/08/01 · openalex publication_date 2014/10/22 · arxiv updated 2014/12/10 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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-L1(m) function in the one-dimensional intermittent maps.