295 research outputs found
Statistics of sums of correlated variables described by a matrix product ansatz
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.Comment: 6 pages, 1 figur
Matrix product representation and synthesis for random vectors: Insight from statistical physics
Inspired from modern out-of-equilibrium statistical physics models, a matrix
product based framework permits the formal definition of random vectors (and
random 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 distributions by a product of matrices of distributions. The
statistical properties stemming from this construction are studied
theoretically: The landscape of the attainable dependence structure is
thoroughly depicted and a stationarity condition for time series is notably
obtained. The remapping of this framework onto that of Hidden Markov Models
enables us to devise an efficient and accurate practical synthesis procedure. A
design procedure is also described permitting the tuning of model parameters to
attain targeted 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 be
actually prescribed.Comment: 10 pages, 4 figures, submitted to IEEE Transactions on Signal
Processin
Matrix products for the synthesis of stationary time series with a priori prescribed joint distributions
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
struc- ture it has under independence while enabling to input con- trolled 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 prop- erties
can be actually prescribed.Comment: 4 pages, 2 figures, conference publication published in IEEE
International Conference on Acoustics, Speech and Signal Processing (ICASSP),
201
Convergence of large deviation estimators
We study the convergence of statistical estimators used in the estimation of
large deviation functions describing the fluctuations of equilibrium,
nonequilibrium, and manmade stochastic systems. We give conditions for the
convergence of these estimators with sample size, based on the boundedness or
unboundedness of the quantity sampled, and discuss how statistical errors
should be defined in different parts of the convergence region. Our results
shed light on previous reports of 'phase transitions' in the statistics of free
energy estimators and establish a general framework for reliably estimating
large deviation functions from simulation and experimental data and identifying
parameter regions where this estimation converges.Comment: 13 pages, 6 figures. v2: corrections focusing the paper on large
deviations; v3: minor corrections, close to published versio
On the existence of a glass transition in a Random Energy Model
We consider a generalized version of the Random Energy Model in which the
energy of each configuration is given by the sum of independent
contributions ("local energies") with finite variances but otherwise arbitrary
statistics. 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 (e.g., if local energies take discrete values), the glass
transition ceases to exist if the number of energy levels grows sufficiently
fast with system size. This shows that statistical independence of energy
levels does not imply the existence of a glass transition.Comment: 12 pages, 2 figures, submitted to J.Phys.
Sommes et extrêmes en physique statistique et traitement du signal (ruptures de convergences, effets de taille finie et représentation matricielle)
Cette thèse s'est développée à l'interface entre physique statistique et traitement statistique du signal, afin d'allier les perspectives de ces deux disciplines sur les problèmes de sommes et maxima de variables aléatoires. Nous avons exploré trois axes d'études qui mènent à s'éloigner des conditions classiques (i.i.d.) : l'importance des événements rares, le couplage avec la taille du système, et la corrélation. Combinés, ces trois axes mènent à des situations dans lesquelles les théorèmes de convergence classiques sont mis en défaut.Pour mieux comprendre l'effet du couplage avec la taille du système, nous avons étudié le comportement de la somme et du maximum de variables aléatoires indépendantes élevées à une puissance dépendante de la taille du signal. Dans le cas du maximum, nous avons mis en évidence l'apparition de lois limites non standards. Dans le cas de la somme, nous nous sommes intéressés au lien entre effet de linéarisation et transition vitreuse en physique statistique. Grâce à ce lien, nous avons pu définir une notion d'ordre critique des moments, montrant que, pour un processus multifractal, celui-ci ne dépend pas de la résolution du signal. Parallèlement, nous avons construit et étudié, théoriquement et numériquement, les performances d'un estimateur de cet ordre critique pour une classe de variables aléatoires indépendantes.Pour mieux cerner l'effet de la corrélation sur le maximum et la somme de variables aléatoires, nous nous sommes inspirés de la physique statistique pour construire une classe de variable aléatoires dont la probabilité jointe peut s'écrire comme un produit de matrices. Après une étude détaillée de ses propriétés statistiques, qui a montré la présence potentielle de corrélation à longue portée, nous avons proposé pour ces variables une méthode de synthèse en réussissant à reformuler le problème en termes de modèles à chaîne de Markov cachée. Enfin, nous concluons sur une analyse en profondeur du comportement limite de leur somme et de leur maximum.This thesis has grown at the interface between statistical physics and signal processing, combining the perspectives of both disciplines to study the issues of sums and maxima of random variables. Three main axes, venturing beyond the classical (i.i.d) conditions, have been explored: The importance of rare events, the coupling between the behavior of individual random variable and the size of the system, and correlation. Together, these three axes have led us to situations where classical convergence theorems are no longer valid.To improve our understanding of the impact of the coupling with the system size, we have studied the behavior of the sum and the maximum of independent random variables raised to a power depending of the size of the signal. In the case of the maximum, we have brought to light non standard limit laws. In the case of the sum, we have studied the link between linearisation effect and glass transition in statistical physics. Following this link, we have defined a critical moment order such that for a multifractal process, this critical order does not depend on the signal resolution. Similarly, a critical moment estimator has been designed and studied theoretically and numerically for a class of independent random variables.To gain some intuition on the impact of correlation on the maximum or sum of random variables, following insights from statistical physics, we have constructed a class of random variables where the joint distribution probability can be expressed as a matrix product. After a detailed study of its statistical properties, showing that these variables can exhibit long range correlations, we have managed to recast this model into the framework of Hidden Markov Chain models, enabling us to design a synthesis procedure. Finally, we conclude by an in-depth study of the limit behavior of the sum and maximum of these random variables.LYON-ENS Sciences (693872304) / SudocSudocFranceF
Renormalization flow for extreme value statistics of random variables raised to a varying power
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 k.q*(n), non-standard limit distributions emerge.Comment: 21 pages, 1 figure,final version, to appear in Journal of Physics
Sums and extremes in statistical physics and signal processing : Convergence breakdowns, finite size effects and matrix representations
Cette thèse s'est développée à l'interface entre physique statistique et traitement statistique du signal, afin d'allier les perspectives de ces deux disciplines sur les problèmes de sommes et maxima de variables aléatoires. Nous avons exploré trois axes d'études qui mènent à s'éloigner des conditions classiques (i.i.d.) : l'importance des événements rares, le couplage avec la taille du système, et la corrélation. Combinés, ces trois axes mènent à des situations dans lesquelles les théorèmes de convergence classiques sont mis en défaut.Pour mieux comprendre l'effet du couplage avec la taille du système, nous avons étudié le comportement de la somme et du maximum de variables aléatoires indépendantes élevées à une puissance dépendante de la taille du signal. Dans le cas du maximum, nous avons mis en évidence l'apparition de lois limites non standards. Dans le cas de la somme, nous nous sommes intéressés au lien entre effet de linéarisation et transition vitreuse en physique statistique. Grâce à ce lien, nous avons pu définir une notion d'ordre critique des moments, montrant que, pour un processus multifractal, celui-ci ne dépend pas de la résolution du signal. Parallèlement, nous avons construit et étudié, théoriquement et numériquement, les performances d'un estimateur de cet ordre critique pour une classe de variables aléatoires indépendantes.Pour mieux cerner l'effet de la corrélation sur le maximum et la somme de variables aléatoires, nous nous sommes inspirés de la physique statistique pour construire une classe de variable aléatoires dont la probabilité jointe peut s'écrire comme un produit de matrices. Après une étude détaillée de ses propriétés statistiques, qui a montré la présence potentielle de corrélation à longue portée, nous avons proposé pour ces variables une méthode de synthèse en réussissant à reformuler le problème en termes de modèles à chaîne de Markov cachée. Enfin, nous concluons sur une analyse en profondeur du comportement limite de leur somme et de leur maximum.This thesis has grown at the interface between statistical physics and signal processing, combining the perspectives of both disciplines to study the issues of sums and maxima of random variables. Three main axes, venturing beyond the classical (i.i.d) conditions, have been explored: The importance of rare events, the coupling between the behavior of individual random variable and the size of the system, and correlation. Together, these three axes have led us to situations where classical convergence theorems are no longer valid.To improve our understanding of the impact of the coupling with the system size, we have studied the behavior of the sum and the maximum of independent random variables raised to a power depending of the size of the signal. In the case of the maximum, we have brought to light non standard limit laws. In the case of the sum, we have studied the link between linearisation effect and glass transition in statistical physics. Following this link, we have defined a critical moment order such that for a multifractal process, this critical order does not depend on the signal resolution. Similarly, a critical moment estimator has been designed and studied theoretically and numerically for a class of independent random variables.To gain some intuition on the impact of correlation on the maximum or sum of random variables, following insights from statistical physics, we have constructed a class of random variables where the joint distribution probability can be expressed as a matrix product. After a detailed study of its statistical properties, showing that these variables can exhibit long range correlations, we have managed to recast this model into the framework of Hidden Markov Chain models, enabling us to design a synthesis procedure. Finally, we conclude by an in-depth study of the limit behavior of the sum and maximum of these random variables
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