Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory

Abstract

Fluctuation geometry was recently proposed as a counterpart approach of Riemannian geometry of inference theory. This theory describes the geometric features of the statistical manifold $\mathcal{M}$ of random events that are described by a family of continuous distributions $dp(x|\theta)$. A main goal of this work is to clarify the statistical relevance of Levi-Civita curvature tensor $R_{ijkl}(x|\theta)$ of the statistical manifold $\mathcal{M}$. For this purpose, the notion of \emph{irreducible statistical correlations} is introduced. Specifically, a distribution $dp(x|\theta)$ exhibits irreducible statistical correlations if every distribution $dp(\check{x}|\theta)$ obtained from $dp(x|\theta)$ by considering a coordinate change $\check{x}=\phi(x)$ cannot be factorized into independent distributions as $dp(\check{x}|\theta)=\prod_{i}dp^{(i)}(\check{x}^{i}|\theta)$. It is shown that the curvature tensor $R_{ijkl}(x|\theta)$ arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar $R(x|\theta)$ allows to introduce a criterium for the applicability of the \emph{gaussian approximation} of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distributions family $dp(x|\theta)$, which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einstein's fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the \emph{invariant fluctuation theorems}.