General limit distributions for sums of random variables with a matrix product representation

TL;DR

This paper characterizes the limit distributions of sums of random variables represented by a matrix product ansatz, linking ergodicity breaking in hidden Markov chains to non-standard distribution behaviors.

Contribution

It introduces a novel approach connecting matrix product representations of random variables to hidden Markov chain ergodicity, providing a comprehensive algorithmic characterization of limit distributions.

Findings

Non-standard limit distributions are derived using a hidden Markov chain framework.

Ergodicity breaking in the underlying chain influences the limit distribution type.

A full algorithmic method for characterizing these distributions is developed.

Abstract

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.