## F. Angeletti, E. Bertin, P. Abry

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

F. Angeletti, E. Bertin, P. Abry, 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.### Random vector and time series definition and synthesis from matrix product representations: From Statistical Physics to Hidden Markov Models,

F. Angeletti, E. Bertin, P. Abry, IEEE Transactions on Signal Processing, (2013), 5389-5400, ### Abstract

Inspired from modern out-of-equilibrium statistical physics models,a matrix product based framework is defined and studied,that permits the formal definition of random vectors and time series whose desired joint distributions are a priori prescribed.
Its key feature consists of preserving the writing of the joint distribution as the simple product structure it has under independence,while inputing controlled dependencies amongst components:
This is obtained by replacing the product of probability densities by a product of matrices of probability densities.
It is first shown that this matrix product model can be remapped onto the framework of Hidden Markov Models.
Second,combining this double perspective enables us both to study the statistical properties of this model in terms of marginal distributions and dependencies (a stationarity condition is notably devised) and to devise an efficient and accurate numerical synthesis procedure.
A design procedure is also described that permits the tuning of model parameters to attain targeted statistical properties.
Pedagogical well-chosen examples of times series and multivariate vectors aim at illustrating the power and versatility of the proposed approach and at showing how targeted statistical properties can actually be prescribed.### Statistics of sums of correlated variables described by a matrix product ansatz,

F. Angeletti, E. Bertin, P. Abry, 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.### Matrix Products for the Synthesis of Stationary Time Series with a priori Prescribed Joint Distributions,

F. Angeletti, E. Bertin, P. Abry, ICASSP conference, (2012), 3897-3900, ### Abstract

Inspired from non-equilibrium statistical physics models,a general framework enabling the definition and synthesis of stationary time series with a priori prescribed and controlled joint distributions is constructed.
Its central feature consists of preserving for the joint distribution the simple product structure it has under independence while enabling to input controlled and prescribed dependencies amongst samples.
To that end,it is based on products of $d$-dimensional matrices,whose entries consist of valid distributions.
The statistical properties of the thus defined time series are studied in details.
Having been able to recast this framework into that of Hidden Markov Models enabled us to obtain an efficient synthesis procedure.
Pedagogical well-chosen examples (time series with the same marginal distribution,same covariance function,but different joint distributions) aim at illustrating the power and potential of the approach and at showing how targeted statistical properties can be actually prescribed.