Related papers: On the reduction of a random basis

Inspired by the idea of Bernoulli decomposition, we give a simple proof for a generalization of Hal\'asz anti--concentration result about random sum of vectores in $\mathbb{R}^d$. From our results, we can give one upper bound for the…

Given a class $\mathcal G$ of graphs, let ${\mathcal G}_n$ denote the set of graphs in $\mathcal G$ on vertex set $[n]$. For certain classes $\mathcal G$, we are interested in the asymptotic behaviour of a random graph $R_n$ sampled…

We analyze the distribution of $\sum_{i=1}^m v_i \bx_i$ where $\bx_1,...,\bx_m$ are fixed vectors from some lattice $\cL \subset \R^n$ (say $\Z^n$) and $v_1,...,v_m$ are chosen independently from a discrete Gaussian distribution over $\Z$.…

For a real $N\ge 1$ and a vector $\xi =(1,\xi_1,...,\xi_n)$ define a matrix $$ {\cal A} (\xi, N) = ({array}{ccccc} N^{-1} & 0& 0& ... &0 \cr N^{\frac{1}{n}} \xi_1 & -N^{\frac{1}{n}} & 0&... & 0 \cr N^{\frac{1}{n}} \xi_2 &0& -N^{\frac{1}{n}}…

Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function…

Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3 \exp(-c\sqrt{n}\r)$ one has $ \sup_{x\in…

We prove that if $\phi: {\Bbb R}^d \times {\Bbb R}^d \to {\Bbb R}$, $d \ge 2$, is a homogeneous function, smooth away from the origin and having non-zero Monge-Ampere determinant away from the origin, then $$ R^{-d} # \{(n,m) \in {\Bbb Z}^d…

The Johnson--Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be flattened to lower dimensions while…

Motivated by a problem arising when analysing data from quarantine searches, we explore properties of distributions of sums of independent means of independent lattice-valued random variables. The aim is to determine the extent to which…

A refinement of so-called fast Johnson-Lindenstrauss transform, due to Ailon and Chazelle (2006), and Matou\v{s}ek (2008), is proposed. While it preserves the time efficiency and simplicity of implementation of the original construction, it…

Given $X$ a random vector in ${\mathbb{R}}^n$, set $X_1,...,X_N$ to be independent copies of $X$ and let $\Gamma=\frac{1}{\sqrt{N}}\sum_{i=1}^N <X_i,\cdot>e_i$ be the matrix whose rows are $\frac{X_1}{\sqrt{N}},\dots, \frac{X_N}{\sqrt{N}}$.…

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$…

We study the spectral properties of a class of random matrices of the form $S_n^{-} = n^{-1}(X_1 X_2^* - X_2 X_1^*)$ where $X_k = \Sigma^{1/2}Z_k$, for $k=1,2$, $Z_k$'s are independent $p\times n$ complex-valued random matrices, and…

We prove that the volumes determined by the lengths of the non-zero vectors $\pm\vecx$ in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the…

We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. This empirical measure is adjusted to match the true probability of sets in a finite partition which changes each…

The aim of the article is to investigate the relative dispersion properties of the Well Mixed class of Lagrangian Stochastic Models. Dimensional analysis shows that given a model in the class, its properties depend solely on a…

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert--Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices we…

We study planar random motions with finite velocities, of norm $c>0$, along orthogonal directions and changing at the instants of occurrence of a non-homogeneous Poisson process with rate function $\lambda(t),\ t\ge0$. We focus on the…

Solutions to the random Fibonacci recurrence x_{n+1}=x_{n} + or - Bx_{n-1} decrease (increase) exponentially, x_{n} = exp(lambda n), for sufficiently small (large) B. In the limits B --> 0 and B --> infinity, we expand the Lyapunov exponent…

We compute exact asymptotic of the statistical density of random matrices belonging to the Generalized Gaussian orthogonal, unitary and symplectic ensembles such that there no eigenvalues in the interval $[\sigma, +\infty[$. In particular,…