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

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 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

Bounds for the covariance of functions of infinite variance stable random variables with applications to central limit theorems and wavelet-based estimation

We establish bounds for the covariance of a large class of functions of infinite variance stable random variables, including unbounded functions such as the power function and the logarithm. These bounds involve measures of dependence between the stable variables, some of which are new. The bounds are also used to deduce the central limit theorem for unbounded functions of stable moving average time series. This result extends the earlier results of Tailen Hsing and the authors on central limit theorems for bounded functions of stable moving averages. It can be used to show asymptotic normality of wavelet-based estimators of the self-similarity parameter in fractional stable motions

Small and large scale behavior of moments of poisson cluster processes

Poisson cluster processes are special point processes that find use in modeling Internet traffic, neural spike trains, computer failure times and other real-life phenomena. The focus of this work is on the various moments and cumulants of Poisson cluster processes, and specifically on their behavior at small and large scales. Under suitable assumptions motivated by the multiscale behavior of Internet traffic, it is shown that all these various quantities satisfy scale free (scaling) relations at both small and large scales. Only some of these relations turn out to carry information about salient model parameters of interest, and consequently can be used in the inference of the scaling behavior of Poisson cluster processes. At large scales, the derived results complement those available in the literature on the distributional convergence of normalized Poisson cluster processes, and also bring forward a more practical interpretation of the so-called slow and fast growth regimes. Finally, the results are applied to a real data trace from Internet traffic.NSA grant [H98230-13-1-0220]info:eu-repo/semantics/publishedVersio

Bootstrap for Multifractal Analysis

Multifractal analysis, which mainly consists in estimating scaling exponents, has become a popular tool for empirical data analysis. Although widely used in different applications, the statistical performance and the reliability of the estimation procedures are still poorly known. Notably, little is known about confidence intervals, though they are of first importance in applications. The present work investigates the potential uses of bootstrap for multifractal estimation: Can bootstrap improve current estimation procedures or be used to obtain reliable confidence intervals~? Comparing the statistical performance of different estimators, our major result is to show that bootstrap based procedures provide us both with accurate estimates and reliable confidence intervals. We believe that this brings substantial improvements to practical empirical multifractal analyses