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Nonlinear Poincar\'e inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon…
For arbitrary two probability measures on real d-space with given means and variances (covariance matrices), we provide lower bounds for their total variation distance. In the one-dimensional case, a tight bound is given.
Derivations of two Bell's inequalities are given in a form appropriate to the interpretation of experimental data for explicit determination of all the correlations. They are arithmetic identities independent of statistical reasoning and…
Assume one observes independent categorical variables or, equivalently, one observes the corresponding multinomial variables. Estimating the distribution of the observed sequence amounts to estimating the expectation of the multinomial…
Sinopoli et al. (TAC, 2004) considered the problem of optimal estimation for linear systems with Gaussian noise and intermittent observations, available according to a Bernoulli arrival process. They showed that there is a "critical"…
We study a version of the Busemann-Petty problem for $\log$-concave measures with an additional assumption on the dilates of convex, symmetric bodies. One of our main tools is an analog of the classical large deviation principle applied to…
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
The properties of the normal distribution under linear transformation, as well the easy way to compute the covariance matrix of marginals and conditionals, offer a unique opportunity to get an insight about several aspects of uncertainties…
There is a natural intuition that, given a large $n$, the distributions of small segments of a randomly sampled polygonal chain and those of a randomly sampled closed polygonal chain (drawn from the subspace measure of course), should be…
Exploiting the geometric nature of statistical divergences, we devise a way to define associated induced uncertainty measures for discrete and finite probability distributions. We also report new uncertainty measures and discuss their…
We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…
We present a class of modified logarithmic Sobolev inequality, interpolating between Poincar\'e and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or more complex $\exp(-|x|^\al\log^\beta(2+|x|))$…
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…
We obtain new oscillation inequalities in metric spaces in terms of the Peetre $K-$functional and the isoperimetric profile. Applications provided include a detailed study of Fractional Sobolev inequalities and the Morrey-Sobolev embedding…
Overlapping coefficient is a direct measure of similarity between two distributions which is recently becoming very useful. This paper investigates estimation for some well-known measures of overlap, namely Matusita's measure $\rho$,…
We obtain in closed form averages of polynomials, taken over hermitian matrices with the Gaussian measure involved in the Kontsevich integral, and prove a conjecture of Witten enabling one to express analogous averages with the full (cubic…
We prove Poincar\'e and Log$^{\beta}$-Sobolev inequalities for probability measures on step-two Carnot groups.
We introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected…
We study the probability that a random polynomial with integer coefficients is reducible when factored over the rational numbers. Using computer-generated data, we investigate a number of different models, including both monic and non-monic…
We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors $w_i$ of a not necessarily orthonormal basis of…