The asymptotic distributional behaviour of transformations preserving infinite measures

The Dynkin-Lamperti arc-sine laws for measure preserving transformations

“…By the standard criterion for regular variation of sequences, {µ(A n )} is slowly varying, in accordance with P (η > 1) = τ (1). This completes the proof of Theorem 1.…”

“…The reasoning in the proof of Proposition 0 in [1] ( cf. also [18]) now shows that for every subsequence of N there exists a subsequence {n k } and a probability measure τ on [0, ∞] such that the sequence {…”

“…if the distribution of X n with respect to ν (ν ∈ D) converges to the distribution of ξ in the usual sense (see [1]). …”

Statistical properties for nonhyperbolic maps with finite range structure

“…Using Lemma 3.6, we can apply Theorem 3.1 in [7] to (T, X, M, T M) so that any A ∈ R(C, T ) ∩ M is a Darling-Kac set for T whose return time process is continued fraction mixing and T is pointwise dual ergodic, i.e., for any f ∈ L 1 (µ) lim n→∞ [2]). Then (3.3-1)-(3.3-2) follow from the results which were established in [1], [4] and [34] (cf. [7]).…”

Distributional behaviors of time-averaged observables in the Langevin equation with fluctuating diffusivity: Normal diffusion but anomalous fluctuations

“…Such anomalous behavior has been studied by infinite ergodic theory in dynamical systems [3]. Infinite ergodic theory states that time-averaged observables converge in distribution, and the distribution function depends on the invariant measure as well as a class of the observation function [4][5][6][7]. Continuous-time random walk (CTRW) is a model of anomalous diffusion, where the mean square displacement (MSD) increases sub-linearly with time, and is extensively studied in disorder materials [8] as well as biophysics [9,10].…”