Self-organized criticality attributed to a central limit-like convergence effect

•We explain self-organized criticality through a convergence effect related to the central limit theorem.•This effect has as its focus of convergence the family of Tweedie exponential dispersion models.•The Tweedie compound Poisson distribution inherently expresses both fluctuation scaling and 1/f1/f noise.•This compound Poisson distribution also can be used to predict the behavior of sandpile models.

Self-organized criticality is a hypothesis used to explain the origin of 1/f1/f noise and other scaling behaviors. Despite being proposed nearly 30 years ago, no consensus exists as to its exact definition or mathematical mechanism(s). Recently, a model for 1/f1/f noise was proposed based on a family of statistical distributions known as the Tweedie exponential dispersion models. These distributions are characterized by an inherent scale invariance that manifests as a variance to mean power law, called fluctuation scaling; they also serve as foci of convergence in a limit theorem on independent and identically distributed distributions. Fluctuation scaling can be modeled by self-similar stochastic processes that relate the variance to mean power law to 1/f1/f noise through their correlation structure. A hypothesis is proposed whereby the effects of self-organized criticality are mathematically modeled by the Tweedie distributions and their convergence behavior as applied to self-similar stochastic processes. Sandpile model fluctuations are shown to manifest 1/f1/f noise, fluctuation scaling, and to conform to the Tweedie compound Poisson distribution. The Tweedie models and their convergence theorem allow for a mechanistic explanation of 1/f1/f noise and fluctuation scaling in phenomena conventionally attributed to self-organized criticality, thus providing a paradigm shift in our understanding of these phenomena.