Related papers: Random vectors in the isotropic position

Consider $n$ i.i.d. random elements on $C[0,1]$. We show that, under an appropriate strengthening of the domain of attraction condition, natural estimators of the extreme-value index, which is now a continuous function, and the normalizing…

Fix $c\in (0,1)$ and let $\Gamma$ be a $\lfloor c n\rfloor$-regular digraph on $n$ vertices drawn uniformly at random. We prove that when $n$ is large, the (non-symmetric) adjacency matrix $M$ of $\Gamma$ is invertible with high…

It is shown that \begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*} for all real $u$, where $U_1,\dots,U_n$ are independent random vectors uniformly distributed on the unit sphere in…

We introduce a natural way to construct a random subset of a homogeneous Cantor set $C$ in $[0,1]$ via random labelings of an infinite $M$-ary tree, where $M\geq 2$. The Cantor set $C$ is the attractor of an equicontractive iterated…

We present a concentration result concerning random weighted projections in high dimensional spaces. As applications, we prove (1) New concentration inequalities for random quadratic forms; (2) The infinity norm of most unit eigenvectors of…

We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma=(\xi_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $\xi$, and set…

A weighted relative isoperimetric inequality in convex cones is obtained via the Monge-Ampere equation. The method improves several inequalities in the literature, e.g. constants in a theorem of Cabre--Ros--Oton--Serra. Applications are…

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and…

We prove an asymptotic Cram\'er's theorem, that is, if the sequence $(X_{n}+ Y_{n})_{n\geq 1}$ converges in law to the standard normal distribution and for every $n\geq 1$ the random variables $X_{n}$ and $Y_{n}$ are independent, then…

Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent identically distributed Rademacher random variables, that is $\mathbb{P}\{\varepsilon_i=\pm1\}=1/2$. Let $S_n=a_1\varepsilon_1+\cdots+a_n\varepsilon_n$, where…

Let K be a convex body in $R^d$. A random polytope is the convex hull $[x_1,...,x_n]$ of finitely many points chosen at random in K. $\Bbb E(K,n)$ is the expectation of the volume of a random polytope of n randomly chosen points. I.…

We consider $n\times n$ random matrices $M_{n}=\sum_{\alpha =1}^{m}{\tau _{\alpha }}\mathbf{y}_{\alpha }\otimes \mathbf{y}_{\alpha }$, where $\tau _{\alpha }\in \mathbb{R}$, $\{\mathbf{y}_{\alpha }\}_{\alpha =1}^{m}$ are i.i.d. isotropic…

We present lower estimates for the best constant appearing in the weak $(1,1)$ maximal inequality in the space $(\R^n,\|\cdot\|_{\iy})$. We show that this constant grows to infinity faster than $(\log n)^{1-o(1)}$ when $n$ tends to…

Consider a sample of a centered random vector with unit covariance matrix. We show that under certain regularity assumptions, and up to a natural scaling, the smallest and the largest eigenvalues of the empirical covariance matrix converge,…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…

Let $X_1, \ldots , X_n$ be i.i.d. random vectors in $\mathbb{R}^d$ with $\|X_1\| \le \beta$. Then, we show that $\frac{1}{\sqrt{n}}(X_1 + \ldots + X_n)$ converges to a Gaussian in quadratic transportation (also known as "Kantorovich" or…

In this paper, we deal with the question; under what conditions the points $P_i(xi,yi)$ $(i = 1,\cdots, n)$ form a convex polygon provided $x_1 < \cdots < x_n$ holds. One of the main findings of the paper can be stated as follows: "Let…

Let $\mathbb{P}_{\kappa}(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $\mathfrak{C}_\kappa$, a regular $\kappa$-gon with area $1$, are in convex position, that is, form the vertex set of a…

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$ H(X_1+\cdots+X_{n+1}) \geq…

The classical problem of maximizing the Shannon entropy of a sum of independent random variables supported on a finite alphabet is considered and settled in the ternary case. Namely, the following theorem is established: if…