Sums and extremes in statistical physics and signal processing

Abstract

Defended on the 6 December 2012, my 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Together, these three axes have led us to situations where classical convergence theorems are no longer valid.
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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.
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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.