Skewed distributions as limits of a formal evolutionary process

TL;DR

This paper demonstrates that skewed and non-normal distributions observed in complex systems can be derived as limits of a formal evolution equation for the probability density function, without requiring detailed knowledge of the underlying dynamics.

Contribution

It introduces a method to derive limiting probability distributions like Gamma, Beta, and Gaussian from a formal evolution equation, bypassing the need for detailed system dynamics.

Findings

The limiting solutions interpolate between common distributions.

The control parameter is the ratio of fluctuation RMS to the range.

No prior knowledge of the system's dynamics is necessary.

Abstract

Time series of observables measured from complex systems do often exhibit non-normal statistics, their statistical distributions (PDF's) are not gaussian and often skewed, with roughly exponential tails. Departure from gaussianity is related to the intermittent development of large-scale coherent structures. The existence of these structures is rooted into the nonlinear dynamical equations obeyed by each system, therefore it is expected that some prior knowledge or guessing of these equations is needed if one wishes to infer the corresponding PDF; conversely, the empirical knowledge of the PDF does provide information about the underlying dynamics. In this work we suggest that it is not always necessary. We show how, under some assumptions, a formal evolution equation for the PDF $p(x)$ can be written down, corresponding to the progressive accumulation of measurements of the generic observable $x$. The limiting solution to this equation is computed analytically, and shown to interpolate between some of the most common distributions, Gamma, Beta and Gaussian PDF's. The control parameter is just the ratio between the rms of the fluctuations and the range of allowed values. Thus, no information about the dynamics is required.