Related papers: On Khinchine type inequalities for pairwise indepe…

Consider some convex body $K\subset\mathbb R^d$. Let $X_1,\dots, X_k$, where $k\leq d$, be random points independently and uniformly chosen in $K$, and let $\xi_k$ be a uniformly distributed random linear $k$-plane. We show that for…

The well-known von Bahr--Esseen bound on the absolute $p$th moments of martingales with $p\in(1,2]$ is extended to a large class of moment functions, and now with a best possible constant factor (which depends on the moment function). This…

Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$…

Suppose that we wish to estimate a vector $\mathbf{x}$ from a set of binary paired comparisons of the form "$\mathbf{x}$ is closer to $\mathbf{p}$ than to $\mathbf{q}$" for various choices of vectors $\mathbf{p}$ and $\mathbf{q}$. The…

This article proposes a new index for quantifying the degree of dependence between random vectors. The index takes values in [0,1] and equals zero if and only if the random vectors are sub-independent. Unlike mere uncorrelatedness,…

For $p\in\lbrack2,\infty]$ a mixed Littlewood-type inequality asserts that there is a constant $C_{(m),p}\geq1$ such that \[ \left( \sum_{i_{1}=1}^{\infty}\left( \sum_{i_{2},...,i_{m}=1}^{\infty }|T(e_{i_{1}},...,e_{i_{m}})|^{2}\right)…

Let \(P_+\) be the Riesz's projection operator and let \(P_-=I-P_+.\) We find the best upper estimates of the expression \(\left\lVert \left( \left\lvert P_+f \right\rvert ^s + \left\lvert P_-f \right\rvert ^s \right) ^{1/s} \right\rVert _p…

We obtain dimension-free concentration inequalities for $\ell^p$-norms, $p\geq2$, of infinitely divisible random vectors with independent coordinates and finite exponential moments. Besides such norms, the methods and results extend to some…

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.

In a prophet inequality problem, $n$ independent random variables are presented to a gambler one by one. The gambler decides when to stop the sequence and obtains the most recent value as reward. We evaluate a stopping rule by the…

We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order p $\in$ [1, $\infty$) between the empirical measure of independent and identically distributed R d-valued random variables…

A {\em maximal inequality} seeks to estimate $\mathbb{E}\max_i X_i$ in terms of properties of the $X_i$. When the latter are independent, the union bound (in its various guises) can yield tight upper bounds. If, however, the $X_i$ are…

We prove scalar and operator-valued Khintchine inequalities for mixtures of free and tensor-independent semicircle variables, interpolating between classical and free Khintchine-type inequalities. Specifically, we characterize the norm of…

To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between…

Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…

We extend some sharp inequalities for martingale-differences to general multiplicative systems of random variables. The key ingredient in the proofs is a technique reducing the general case to the case of Rademacher random variables without…

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…

The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer…

Let $(X_i)_{i=1,...,n}$ be a possibly nonstationary sequence such that $\mathscr{L}(X_i)=P_n$ if $i\leq n\theta$ and $\mathscr{L}(X_i)=Q_n$ if $i>n\theta$, where $0<\theta <1$ is the location of the change-point to be estimated. We…

We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an…