Showing papers by "Amir Dembo published in 2009"

Spectral Measure of Heavy Tailed Band and Covariance Random Matrices

Abstract: We study the asymptotic behavior of the appropriately scaled and possibly perturbed spectral measure $${{\hat{\mu}}}$$ of large random real symmetric matrices with heavy tailed entries. Specifically, consider the N × N symmetric matrix $${{\bf Y}_N^\sigma}$$ whose (i, j) entry is $${\sigma\left(\frac{i}{N}, \frac{j}{N}\right) x_{ij}}$$ , where (x ij , 1 ≤ i ≤ j < ∞) is an infinite array of i.i.d real variables with common distribution in the domain of attraction of an α-stable law, $${\alpha\in (0,2)}$$ , and σ is a deterministic function. For random diagonal D N independent of $${{\bf Y}_N^\sigma}$$ and with appropriate rescaling a N , we prove that $${{\hat{\mu}}_{a_N^{-1} {\bf Y}_N^\sigma + {\varvec {D}}_N}}$$ converges in mean towards a limiting probability measure which we characterize. As a special case, we derive and analyze the almost sure limiting spectral density for empirical covariance matrices with heavy tailed entries.

LDP for Finite Dimensional Spaces

Abstract: This chapter is devoted to the study of the LDP in a framework that is not yet encumbered with technical details. The main example studied is the empirical mean of a sequence of random variables taking values in ℝ d . The concreteness of this situation enables the LDP to be obtained under conditions that are much weaker than those that will be imposed in the “general” theory. Many of the results presented here have counterparts in the infinite dimensional context dealt with later, starting in Chapter 4.

Gibbs Measures and Phase Transitions on Sparse Random Graphs

Abstract: Many problems of interest in computer science and information theory can be phrased in terms of a probability distribution over discrete variables associated to the vertices of a large (but finite) sparse graph. In recent years, considerable progress has been achieved by viewing these distributions as Gibbs measures and applying to their study heuristic tools from statistical physics. We review this approach and provide some results towards a rigorous treatment of these problems.