Related papers: Sharp Khinchin-type inequalities for symmetric dis…
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.
This article investigates sharp comparison of moments for various classes of random variables appearing in a geometric context. In the first part of our work we find the optimal constants in the Khintchine inequality for random vectors…
We sharpen the moment comparison inequalities with sharp constants for sums of random vectors uniform on Euclidean spheres, providing a deficit term (optimal in high dimensions).
We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands are…
We prove a sharp upper bound on negative moments of sums of independent Steinhaus random variables (that is uniform on circles in the plane). Together with the series of earlier works: K\"onig-Kwapie\'n (2001), Baernstein II-Culverhouse…
A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures…
We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by…
We compute the best constant in the Khintchine inequality under assumption that the sum of Rademacher random variables is zero.
Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration…
We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ $k$-wise independent Rademacher random variables. We show that an analogue of Khintchine's inequality holds, with a constant $N^{1/2-k/2p}$, when $k$ is even.…
The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of…
We give a strengthening of the classical Khintchine inequality between the second and the $p$-th moment for $p \ge 3$ with optimal constant by adding a deficit depending on the vector of coefficients of the Rademacher sum.
We consider Khintchine type inequalities on the $p$-th moments of vectors of $N$ pairwise independent Rademacher random variables. We establish that an analogue of Khintchine's inequality cannot hold in this setting with a constant that is…
For discrete martingale-difference sequences $d=\{d_1,\ldots,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak S (d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove…
In this paper we show that the subset of integers that satisfies the Khintchine inequality for $p=1$ with the optimal constant ${\sqrt{2}}$ has to be a $Z_2$ set. We further prove a similar result for a large class of discrete groups. Our…
In this paper we obtain the optimal constants of some classical inequalities, such as the multiple Khinchine inequality for Steinhaus variables and the mixed Littlewood inequality for complex scalars.
To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is…
In this article, we show that a linear combination $X$ of $n$ independent, unbiased Bernoulli random variables $\{X_k\}$ can match the first $2n$ moments of a random variable $Y$ which is uniform on an interval. More generally, for each $p…
We prove limit relations between the sharp constants in the multivariate Bernstein-Nikolskii type inequalities for trigonometric polynomials and entire functions of exponential type with the spectrum in a centrally symmetric convex body.
The current status concerning Hardy-type inequalities with sharp constants is presented and described in a unified convexity way. In particular, it is then natural to replace the Lebesgue measure $dx$ with the Haar measure $dx/x.$ There are…