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

¹ Université de Lyon, Laboratoire de Physique, ENS Lyon, CNRS - 46 Allée d'Italie, F-69007 Lyon, France
² National Institute for Theoretical Physics (NITheP) - Stellenbosch 7600, South Africa
³ Institute of Theoretical Physics, University of Stellenbosch - Stellenbosch 7600, South Africa
⁴ Laboratoire Interdisciplinaire de Physique, Université Joseph Fourier Grenoble, CNRS UMR 5588 BP 87, F-38402 Saint - Martin d'Hères, France

Received: 17 September 2013
Accepted: 6 December 2013

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 in which the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In contrast, when the correlation extends over the 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 non-standard 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.