Distributional Behavior of Time Averages of Non-L-1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

摘要

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of 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- functions. Here, we provide another distributional behavior of time averages of non- functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non- functions which vanish at the indifferent fixed point. We call this class of observation functions weak non- 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- function in the one-dimensional intermittent maps.