Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length

In this paper, we study ergodic properties of $\alpha$-stable autoregressive fractionally integrated moving average ($\text{ARFIMA}$) 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 $F\left(n\right)$ for the dynamical functional $D\left(n\right)$ defined as a Fourier transform of the $n$-lag increments of the $\text{ARFIMA}$ process. Moreover, we introduce more flexible concept of the $\epsilon$-ergodicity.

Figure 1

The test functional $F_{\alpha }\left(n\right)$ for ARFIMA(1,$d$,1) model with various values of $\alpha , d, \varphi$, and $\theta$. The blue lines are the values obtained from 1000 simulated sample paths, each of the length 500, while the red ones are the theoretical ones; the green dashed lines are 95% confidence intervals. Please note that the imaginary part is theoretically equal to 0, as according to the formula (9); during the simulations some numerical noise occurs.

Figure 2

The minimal sample length $n_0\left(\varepsilon \right)$ of $\text{ARFIMA}(0,-0.11,0)$ process for different $\alpha$-stable innovations. Please note that the $OY$ axis is in log scale to better present the common behavior.

Figure 3

The test functional $F_{\alpha }\left(n\right)$ for $\alpha =1$ with respect to $n$ for few $d$ values.

Figure 4

The test functional $F_{\alpha }\left(n\right)$ for $\alpha =2$ with respect to $n$ for few $d$ values.