Large deviations for occupation and waiting times of infinite ergodic transformations
Toru Sera
TL;DR
The paper develops a unified large-deviation theory for occupation and waiting times in infinite-measure dynamical systems, tying together Dynkin--Lamperti, Darling--Kac, and Lamperti arcsine-type laws under a common set of regular-variation and uniform-sweeping hypotheses. By employing a double Laplace transform approach, it yields explicit asymptotics for quantities like $\mu_H(Z^Y_n/n\le c(n))$ and $\mu_{1_Y}(S^Y_n/a(n)\le c(n))$, with tails governed by $\alpha\in(0,1)$ and slowly varying functions. The results are then extended to multi-set settings $A_i$ (Lamperti-type laws) and applied to Thaler's intermittent maps, demonstrating sharp large-deviation estimates for systems with infinite invariant measures. This framework provides a rigorous bridge between renewal-type limit theorems in infinite ergodic theory and precise large-deviation tails, with potential consequences for the study of intermittent dynamics and related stochastic processes.
Abstract
We establish large deviation estimates related to the Darling--Kac theorem and generalized arcsine laws for occupation and waiting times of ergodic transformations preserving an infinite measure, such as non-uniformly expanding interval maps with indifferent fixed points. For the proof, we imitate the study of generalized arcsine laws for occupation times of one-dimensional diffusion processes and adopt a method of double Laplace transform.