Related papers: Variance bounds, with an application to norm bound…

For two $n \times n$ complex matrices $A$ and $B$, we define the $q$-deformed commutator as $[ A, B ]_q := A B - q BA$ for a real parameter $q$. In this paper, we investigate a generalization of the B\"{o}ttcher-Wenzel inequality which…

We propose a consistent estimator of sharp bounds on the variance of the difference-in-means estimator in completely randomized experiments. Generalizing Robins [Stat. Med. 7 (1988) 773-785], our results resolve a well-known identification…

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is…

By employing a weighted Frobenius norm with a positive matrix $\omega$, we introduce natural generalizations of the famous B\"ottcher-Wenzel (BW) inequality. Based on the combination of the weighted Frobenius norm $\|A\|_\omega :=…

Leveraging tools from convex analysis and incorporating additional singular value information of matrices, we completely resolve the problem of establishing perturbation bounds for the Frobenius norm of subunitary and positive polar…

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csisz\'ar and Talata. It is…

We introduce a smooth variance sum associated to a pair of positive definite symmetric integral matrices $A_{m\times m}$ and $B_{n\times n}$, where $m\geq n$. By using the oscillator representation, we give a formula for this variance sum…

For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i…

We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds…

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field, and that of a normal vector with a positive definite covariance matrix. Our bounds are commensurate to the…

We give lower bounds on the largest singular value of arbitrary matrices, some of which are asymptotically tight for almost all matrices. To study when these bounds are exact, we introduce several combinatorial concepts. In particular, we…

This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulk and at the edge of the spectrum, as well as for some intermediate eigenvalues. Relying on the GUE example,…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

We prove asymptotic lower bounds on the variance of the number of vertices and missed area of random disc-polygons in convex discs whose boundary is $C_+^2$ smooth. The established lower bounds are of the same order as the upper bounds…

Let $A\in\mathbb{C}^{n\times n}$ be a normal matrix with spectrum $\{\lambda_{i}\}_{i=1}^{n}$, and let $\widetilde{A}=A+E\in\mathbb{C}^{n\times n}$ be a perturbed matrix with spectrum $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$. If…

It is shown that at least 50% of the probability mass of a sum of independent Rademacher random variables is within one standard deviation from its mean. This lower bound is sharp, it is much better than for instance the bound that can be…

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that…

We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix…

We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of…

We give tight upper bounds on the numerical range of square matrices in terms of their Frobenius (Euclidian) norm and a combinatorial parameter similar to the clique number of graphs. Our results imply a concise form of the fundamental…