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.