  Canonical characterization techniques that rely upon mean squared
displacement ($\mathrm{MSD}$) break down for non-ergodic processes, making it
challenging to characterize anomalous diffusion from an individual time-series
measurement. Non-ergodicity reigns when the time-averaged mean square
displacement $\mathrm{TA}$-$\mathrm{MSD}$ differs from the ensemble-averaged
mean squared displacement $\mathrm{EA}$-$\mathrm{MSD}$ even in the limit of
long measurement series. In these cases, the typical theoretical results for
ensemble averages cannot be used to understand and interpret data acquired from
time averages. The difficulty then lies in obtaining statistical descriptors of
the measured diffusion process that are not non-ergodic. We show that linear
descriptors such as the standard deviation ($SD$), coefficient of variation
($CV$), and root mean square ($RMS$) break ergodicity in proportion to
non-ergodicity in the diffusion process. In contrast, time series of
descriptors addressing sequential structure and its potential nonlinearity:
multifractality change in a time-independent way and fulfill the ergodic
assumption, largely independent of the time series' non-ergodicity. We show
that these findings follow the multiplicative cascades underlying these
diffusion processes. Adding fractal and multifractal descriptors to typical
linear descriptors would improve the characterization of anomalous diffusion
processes. Two particular points bear emphasis here. First, as an appropriate
formalism for encoding the nonlinearity that might generate non-ergodicity,
multifractal modeling offers descriptors that can behave ergodically enough to
meet the needs of linear modeling. Second, this capacity to describe
non-ergodic processes in ergodic terms offers the possibility that multifractal
modeling could unify several disparate non-ergodic diffusion processes into a
common framework.
