Distributional laws in infinite ergodic theory

Abstract

Conservative ergodic measure preserving transformations on infinite measure spaces are considered. The asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition are investigated. As applications we derive asymptotic laws for the normalized Kac process and the normalized spent time Kac process. The notion of uniformly returning sets is introduced. For these sets it is proven that if the wandering rate is slowly varying then the normalized spent time Kac process converges strongly distributional to a random variable uniformly distributed on the unit interval.As a number theoretical application the continued fraction digits are considered as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable uniformly distributed on the unit interval. For this process normalized linearly a large deviation asymptotic is proved