title: Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length.
abstract: In this paper, we study ergodic properties of α-stable autoregressive fractionally integrated moving average (ARFIMA) processes which form a large class of anomalous diffusions. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim that the analyzed data are ergodic or not. This will be solved by checking the asymptotic convergence to 0 of the empirical estimator F(n) for the dynamical functional D(n) defined as a Fourier transform of the n-lag increments of the ARFIMA process. Moreover, we introduce more flexible concept of the ε-ergodicity.
authors.affiliation: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland.
authors.forename: Hanna
authors.initials: H
authors.lastname: Loch-Olszewska
authors.affiliation: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland.
authors.forename: Grzegorz
authors.initials: G
authors.lastname: Sikora
authors.affiliation: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland.
authors.forename: Joanna
authors.initials: J
authors.lastname: Janczura
authors.affiliation: Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, 50-370 Wrocław, Poland.
authors.forename: Aleksander
authors.initials: A
authors.lastname: Weron