Title: Anomalous diffusion ɛ -ergodicity testing approach
Author: Hanna Loch-Olszewska
The ergodicity is a very important feature for the real-life processes. Together with the stationarity it guarantees satisfying the Bolzmann hypothesis,
i.e. that the temporal and ensemble averages coincide. Hence, for an ergodic process, the observation of many realizations in one time point can be equivalent to observing single trajectory over a long time horizon. The presence of ergodicity in a sample from a random process can be veried using the empirical estimator F(n) for the dynamical functional D(n), dened as a Fourier transform of the n-lag increments of the process. Lately, in various elds of physics, biology and related sciences, the non-ergodic systems that show the anomalous behaviour have been noted. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim the ergodicity breaking of the sample. Using empirical
simulations of the functional or computations of theoretical formula for different values of parameters of the process, one can verify so-called “-e-ergodicity the convergence of the estimator to some predened thin interval. The current work is focused on autoregressive fractionally integrated moving average (ARFIMA) processes with different innovation processes, which form a large class of anomalous diffusions.