Statistical physics and signal processing in interaction
Interactions between tools and concepts of Statistical Physics and Signal Processing have been explored to see how it can help to analyze and understand situations and models (inspired from Statistical Physics) where usual convergence theorems fail. Studies of independent random variables raised to a power depending on the sample size were shown to yield non standard limit distributions for the maximum. For sums, it provided a link between linearization effect in moment estimation and glass transition in statistical physics. In addition, it formalized the existence of an intrinsic critical moment order for a multifractal process, thus comforting earlier results. A critical moment estimator has been defined and studied for a class of independent (yet with intricate marginal dis- tribution) random variables. A class of random variables with intricate correlation has been studied, whose joint distributions is written as a product of matrices and which can have long range correlations. This model can also be recast into the framework of Hidden Markov Chain models, leading to theoretical design and actual synthesis. The limit behavior of the sum of such random variables has been characterized, both using rescaled limit distributions and large deviations.