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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.

Large deviations for correlated random variables described by a matrix product ansatz,

, , , , JSTAT, (2014), 2003,

Abstract

We study the large deviations of sums of correlated random variables described by a matrix product ansatz,which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of correlations. We show with specific examples that different large deviation behaviors can be found with this ansatz. In particular,it is possible to construct sums of correlated random variables that violate the Law of Large Numbers,the Central Limit Theorem,as well as sums that have nonconvex rate functions or rate functions with linear parts or plateaux.

2013

On the existence of a glass transition in a Random Energy Model,

, , , J. Phys. A, (2013), 315002,

Abstract

We revisit the Random Energy Model by assuming that the energy of each configuration is given by the sum of Nindependent contributions (local energies'') with finite variances,instead of directly assuming a Gaussian energy distribution with a variance proportional to N. Using the large deviation formalism,we find that the glass transition generically exists when local energies have a smooth distribution. In contrast,if the distribution of the local energies has a Dirac mass at the minimal energy,the glass transition may cease to exist. This property is illustrated on a simple example,in which local energies are equal to zero with a finite probability,and positive otherwise.

Random vector and time series definition and synthesis from matrix product representations: From Statistical Physics to Hidden Markov Models,

, , , 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,

, , , 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.

2012

Renormalization flow for extreme value statistics of random variables raised to a varying power,

, , , J. Phys. A, (2012), 115004,

Abstract

Using a renormalization approach,we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power $q_n$ that varies monotonically with the sample size $n$. Under these conditions,a non-standard class of max-stable limit distributions,which mirror the classical ones,emerges. Furthermore a transition mechanism between the classical and the non-standard limit distributions is brought to light. If $q_n$ grows slower than a characteristic function $q^*_n$,the standard limit distributions are recovered,while if $q_n$ behaves asymptotically as $\lambda q^*_n$,non-standard limit distributions emerge.

Matrix Products for the Synthesis of Stationary Time Series with a priori Prescribed Joint Distributions,

, , , 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.

Critical moment definition and estimation,for finite size observation of log-exponential-power law random variables,

, , , Signal Processing, (2012), 2848-2865,

Abstract

This contribution aims at studying the behaviour of the classical sample moment estimator,$S(n,q)=\sum_k=1^n X_k^q/n$,as a function of the number of available samples $n$,in the case where the random variables $X$ are positive,have finite moments at all orders and are naturally of the form $X=\exp Y$ with the tail of $Y$ behaving like $\exp(-y^\rho)$. This class of laws encompasses and generalizes the classical example of the log-normal law. This form is motivated by a number of applications stemming from modern statistical physics or multifractal analysis. Borrowing heuristic and analytical results from the analysis of the Random Energy Model in statistical physics,a critical moment $q_c(n)$ is defined as the largest statistical order $q$ up to which the sample mean estimator $S(n,q)$ correctly accounts for the ensemble average $\E X^q$,for a given $n$. A practical estimator for the critical moment $q_c(n)$ is then proposed. Its statistical performance are studied analytically and illustrated numerically in the case of i.i.d. samples. A simple modification is proposed to explicitly account for correlation amongst the observed samples. Estimation performance are then carefully evaluated by means of Monte-Carlo simulations in the practical case of correlated time series.

2011

Linearization effect in multifractal analysis: Insights from Random Energy Model analysis,

, , , , Physica D, (2011), 1245-1253,

Abstract

The analysis of the linearization effect in multifractal analysis,and hence of the estimation of moments for multifractal processes,is revisited borrowing concepts from the statistical physics of disordered systems,notably from the analysis of the so-called Random Energy Model. Considering a standard multifractal process (compound Poisson motion),chosen as a simple representative example,we show: i) the existence of a critical order $q^*$ beyond which moments,though finite,cannot be estimated through empirical averages,irrespective of the sample size of the observation; ii) that multifractal exponents necessarily behave linearly in $q$,for $q > q^*$. Tayloring the analysis conducted for the Random Energy Model to that of Compound Poisson motion,we provide explicative and quantitative predictions for the values of $q^*$ and for the slope controlling the linear behavior of the multifractal exponents. These quantities are shown to be related only to the definition of the multifractal process and not to depend on the sample size of the observation. Monte-Carlo simulations,conducted over a large number of large sample size realizations of compound Poisson motion,comfort and extend these analyses.