Statistics of sums of correlated variables described by a matrix product ansatz

TL;DR

This paper analyzes the asymptotic distribution of sums of correlated variables modeled by a matrix product ansatz, revealing Gaussian or nonstandard limit laws depending on correlation length, with implications for statistical physics.

Contribution

It characterizes the limiting distributions of sums of correlated variables using a matrix product ansatz, including nonstandard laws when correlations are extensive.

Findings

Finite correlation length leads to Gaussian distribution.

Infinite correlation length causes nonstandard limit distributions.

Connections established with statistical physics models.

Abstract

We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases when the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In constrast, when correlation extends over system size, we observe either a breaking of the law of large numbers, with the onset of giant fluctuations, or a generalization of the central limit theorem with a family of nonstandard limit distributions. The corresponding distributions are found as mixtures of delta functions for the generalized law of large numbers, and as mixtures of Gaussian distributions for the generalized central limit theorem. Connections with statistical physics models are emphasized.