Multifractal descriptors ergodically characterize non-ergodic multiplicative cascade processes

Biological and psychological processes routinely break ergodicity, meaning they fail to have stable means (Mean) and independent variation over time that we might find in additive white Gaussian noise (awGn). One possible reason for this failure of ergodicity is the failure of biological and psychological processes to exhibit independence across time. Multifractal evidence has long suggested that biological and psychological processes show strong signatures of nonlinear interactions across scales. These cross-scale interactions sooner befit a cascade-dynamical process than awGn. The present work thus compares awGn to simulations of multiplicative binomial cascades, submitting both types of series and shuffled versions of each to the Thirumalai-Mountain method for estimating ergodicity breaking. Estimating ergodicity breaking for original awGn and cascade series allows us to examine the sources of ergodicity breaking across the sequence, e.g., in temporal correlations specifying nonlinear interactions across scales, and examining ergodicity breaking of the shuffled series allows us to assess the raw, sequence-independent contribution of distributional properties (e.g., the heavy tails of a cascade) without the original temporal sequence. Raw cascade fluctuations and the standard deviation (SD) and root mean square (RMS) series describing those raw fluctuations break ergodicity, but nonlinear, cascade-dynamical descriptors: multifractal spectrum width (Δα) and multifractal nonlinearity (tMF), maintain ergodicity. Interestingly, the fundamentally linear descriptor, fractal Hurst exponent (HfGn) shows moderate ergodicity breaking when describing the fundamentally nonlinear cascade processes, but the linear descriptor coefficient of variation (CV) controls for multiplicative relationships between SD and Mean and maintains ergodicity. We conclude that the ergodicity of statistical descriptors depends on how well they can portray nonlinearity (Δα and tMF) or at least multiplicativity (CV) of the underlying cascade processes.