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2014

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

, , , Journal of Statistical Physics, (2014), 1255-1283,

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.

2013

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

, , , European Physics Letters, (2013), 50009,

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.