Skewed distributions as limits of a formal evolutionary process

Abstract

Time series of observables measured from different turbulent systems do often exhibit non-normal statistics, their probability distribution functions (PDFs) are not Gaussian and often skewed, with roughly exponential tails, skewness and kurtosis. 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 advance the suggestion that this is not always necessary. We show that, 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 PDFs. The control parameter is just the ratio between the root mean square of the fluctuations and the range of allowed values. Thus, no information about the dynamics is required.