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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. [ABSTRACT FROM AUTHOR]

We study a scenario under which variable step random walks give anomalous statistics. We begin by analyzing the Martingale Central Limit Theorem to find a sufficient condition for the limit distribution to be non-Gaussian. We study the case when the scaling index∼ζ is∼12. For corresponding continuous time processes, it is shown that the probability density function W( x; t) satisfies the Fokker–Planck equation. Possible forms for the diffusion coefficient are given, and related to W( x, t). Finally, we show how a time-series can be used to distinguish between these variable diffusion processes and Lévy dynamics. [ABSTRACT FROM AUTHOR]