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The general limit distributions of the sum of random variables described by afinite matrix product ansatz are characterized. Using a mapping to a HiddenMarkov Chain formalism, non-standard limit distributions are obtained, andrelated to a form of ergodicity breaking in the underlying non-homogeneousHidden Markov Chain. The link between ergodicity and limit distributions isdetailed and used to provide a full algorithmic characterization of the generallimit distributions.

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