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We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding $ell_2$-operator norms. The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.
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We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.
We study the singularity probability of random integer matrices. Concretely, the probability that a random $n times n$ matrix, with integer entries chosen uniformly from ${-m,ldots,m}$, is singular. This problem has been well studied in two regimes: large $n$ and constant $m$; or large $m$ and constant $n$. In this paper, we extend previous techniques to handle the regime where both $n,m$ are large. We show that the probability that such a matrix is singular is $m^{-cn}$ for some absolute constant $c>0$. We also provide some connections of our result to coding theory.
For an $ntimes n$ matrix $A_n$, the $rto p$ operator norm is defined as $$|A_n|_{rto p}:= sup_{boldsymbol{x} in mathbb{R}^n:|boldsymbol{x}|_rleq 1 } |A_nboldsymbol{x}|_pquadtext{for}quad r,pgeq 1.$$ For different choices of $r$ and $p$, this norm corresponds to key quantities that arise in diverse applications including matrix condition number estimation, clustering of data, and finding oblivious routing schemes in transportation networks. This article considers $rto p$ norms of symmetric random matrices with nonnegative entries, including adjacency matrices of ErdH{o}s-Renyi random graphs, matrices with positive sub-Gaussian entries, and certain sparse matrices. For $1< pleq r< infty$, the asymptotic normality, as $ntoinfty$, of the appropriately centered and scaled norm $|A_n|_{rto p}$ is established. When $p geq 2$, this is shown to imply, as a corollary, asymptotic normality of the solution to the $ell_p$ quadratic maximization problem, also known as the $ell_p$ Grothendieck problem. Furthermore, a sharp $ell_infty$-approximation bound for the unique maximizing vector in the definition of $|A_n|_{rto p}$ is obtained. This result, which may be of independent interest, is in fact shown to hold for a broad class of deterministic sequences of matrices having certain asymptotic expansion properties. The results obtained can be viewed as a generalization of the seminal results of F{u}redi and Koml{o}s (1981) on asymptotic normality of the largest singular value of a class of symmetric random matrices, which corresponds to the special case $r=p=2$ considered here. In the general case with $1< pleq r < infty$, spectral methods are no longer applicable, and so a new approach is developed, which involves a refined convergence analysis of a nonlinear power method and a perturbation bound on the maximizing vector.
Gibbs-type random probability measures and the exchangeable random partitions they induce represent an important framework both from a theoretical and applied point of view. In the present paper, motivated by species sampling problems, we investigate some properties concerning the conditional distribution of the number of blocks with a certain frequency generated by Gibbs-type random partitions. The general results are then specialized to three noteworthy examples yielding completely explicit expressions of their distributions, moments and asymptotic behaviors. Such expressions can be interpreted as Bayesian nonparametric estimators of the rare species variety and their performance is tested on some real genomic data.
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.
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