### Matrix-correlated random variables: a statistical physics and signal processing dialogue,

F. Angeletti, Séminaire LPT Toulouse, Toulouse, (2015), ### Abstract

Finding statistical descriptions of non-equilibrium stationary state is often an arduous task. For specific systems,like ASEP or 1D diffusion-reaction systems,stationary solutions have been characterized using matrix product representations. These representations generalize the product structure of independent random variable to matrices; the non-commutativity of matrices generates correlation while preserving many of the algebraic properties of expectation. Depending on the matrix structure,the correlation can vary from short-range to long-range correlation. Moreover,from a signal processing perspective,these matrix-correlated random variables can be recast as specific Hidden Markov Models. In this talk,we propose to investigate the general statistical properties of this mathematical framework,with the long-term hope to improve our understanding of related physical systems. In particular,we shall focus on the statistical properties of sums of such random variables. Do we have a large deviation principle for this sum? Can we find analogous of the law of large number or the central limit theorem?
### Matrix-correlated random variables: a statistical physics and signal processing dialogue,

F. Angeletti, Séminaire LPTMS Orsay, Orsay, (2015), ### Abstract

Finding statistical descriptions of non-equilibrium stationary state is often an arduous task. For specific systems,like ASEP or 1D diffusion-reaction systems,stationary solutions have been characterized using matrix product representations. These representations generalize the product structure of independent random variable to matrices; the non-commutativity of matrices generates correlation while preserving many of the algebraic properties of expectation. Depending on the matrix structure,the correlation can vary from short-range to long-range correlation. Moreover,from a signal processing perspective,these matrix-correlated random variables can be recast as specific Hidden Markov Models. In this talk,we propose to investigate the general statistical properties of this mathematical framework,with the long-term hope to improve our understanding of related physical systems. In particular,we shall focus on the statistical properties of sums of such random variables. Do we have a large deviation principle for this sum? Can we find analogous of the law of large number or the central limit theorem?
### Critical order for moment estimation : insights from statistical physics.,

F. Angeletti, Club journal LPT Orsay, Orsay, (2015), ### Abstract

Moment estimation is one of the most basic question in statistical signal processing.
For i.i.d. random variables and a infinite number of observations,the law of large numbers implies that the classical moment estimator converges towards the theoretical moment. However,these hypothesis are not really satisfied in practical settings where only a finite number of observations is available and most data are correlated.
It is therefore quite natural to ask ourselves: up to which order can moments be reliably estimated from a given number of observation? In this presentation,I will show how the notion of glassy phase transition developed for studying disordered systems in statistical physics can help to answer such a question.