Nonlinear statistical coupling

TL;DR

This paper explores nonlinear statistical coupling using Tsallis statistics, interpreting escort probabilities as coupled probabilities, and extends the framework to analyze non-stationary stochastic processes with additive and multiplicative noise.

Contribution

It introduces a conjugate transformation linking heavy-tail and compact-support coupled-Gaussians, extending the generalized Fourier transform and analyzing noise effects in nonlinear coupling.

Findings

Coupled-Gaussian distributions depend on the nonlinear coupling parameter Q.

The conjugate transformation maps heavy-tail to compact-support distributions.

Additive and multiplicative noise effects are separable via coupled-variance and Q.

Abstract

By considering a nonlinear combination of the probabilities of a system, a physical interpretation of Tsallis statistics as representing the nonlinear coupling or decoupling of statistical states is proposed. The escort probability is interpreted as the coupled probability, with Q = 1 - q defined as the degree of nonlinear coupling between the statistical states. Positive values of Q have coupled statistical states, a larger entropy metric, and a maximum coupled-entropy distribution of compact-support coupled-Gaussians. Negative values of Q have decoupled statistical states and for -2 < Q < 0 a maximum coupled-entropy distribution of heavy-tail coupled-Gaussians. The conjugate transformation between the heavy-tail and compact-support domains is shown to be -2Q/(2+Q) for coupled-Gaussian distributions. This conjugate relationship has been used to extend the generalized Fourier transform to the compact-support domain and to define a scale-invariant correlation structure with heavy-tail limit distribution. In the present paper, we show that the conjugate is a mapping between the source of nonlinearity in non-stationary stochastic processes and the nonlinear coupling which defines the coupled-Gaussian limit distribution. The effects of additive and multiplicative noise are shown to be separable into the coupled-variance and the coupling parameter Q, providing further evidence of the importance of the generalized moments.