Related papers: Best constants in Rosenthal-type inequalities and …

Let $X$ be a symmetric quasi-Banach function space with Fatou property and let $E$ be an arbitrary symmetric quasi-Banach sequence space. Suppose that $(f_k)_{k\geq0}\subset X$ is a sequence of independent random variables. We present a…

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…

In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,\mu,\sigma)$ and $0\leq a\leq1$:…

We establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers.…

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 give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in $1$-dimensional functional inequalities.

This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special…

In this paper we prove multilevel concentration inequalities for bounded functionals $f = f(X_1, \ldots, X_n)$ of random variables $X_1, \ldots, X_n$ that are either independent or satisfy certain logarithmic Sobolev inequalities. The…

Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space, $\xi$ be a random variable on $(\Omega, \mathcal{F}, \mathbf{P})$, $\mathcal{G}$ be a sub-$\sigma$-algebra of $\mathcal{F}$, and let $\mathbf{E}^\mathcal{G} = \mathbf{ E}(\cdot…

Let $X$ be a Banach space and let $(\xi_j)_{j\ge 1}$ be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions are equivalent: (1). There exists a constant $K$ such that $$…

Let $X$ be a ball Banach function space on $\mathbb{R}^n$, $k\in\mathbb{N}$, $h\in\mathbb{R}^n$, and $\Delta^k_h$ denote the $k${\rm th} order difference. In this article, under some mild extra assumptions about $X$, the authors prove that,…

We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $(S(t,s))_{0\leq s\leq T}$ is a $C_0$-evolution family of contractions on a…

Let $X$ be a random variable and define its concentration function by $$\mathcal{Q}_{h}(X)=\sup_{x\in \mathbb{R}}\mathbb{P}(X\in (x,x+h]).$$ For a sum $S_n=X_1+\cdots+X_n$ of independent real-valued random variables the Kolmogorov-Rogozin…

Let $M,N$ be real-valued martingales such that $N$ is differentially subordinate to $M$. The paper contains the proofs of the following weak-type inequalities: (i) If $M\geq0$ and $0<p\leq1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert…

We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their…

~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let $\{X, X_{n}; n \geq…

We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ together with a sequence of independent, identically distributed $X$-space valued random variables $\xi_1,\dots,\xi_n$ and give a good estimate on the…

The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the…

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm \(\|X\|_{\psi_2}\) and the sub-Gaussian parameter \(\sigma_X\) for centered real-valued random variables. We show that \(\sqrt{3/8} \cdot…

Let $V$ be a symmetric convex body in $\R^m$. We prove sharp Bernstein-type inequalities for entire functions of exponential type with the spectrum in $V$ and discuss certain properties of the extremal functions. Markov-type inequalities…