相关论文: Moments of the Cram\'er transform of log-concave p…

We study the maximal perimeter constant of isotropic log-concave probability measures on $\mathbb{R}^n$. For a measure $\mu$, this quantity, denoted by $\Gamma(\mu)$, is defined as the supremum of the $\mu$-perimeter over all convex bodies…

Let $\mu$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed…

Let $\lambda$ be a probability measure on $\mathbb T^{n-1}$ where $n=2$ or 3. Suppose $\lambda$ is invariant, ergodic and has positive entropy with respect to the linear transformation defined by a hyperbolic matrix. We get a measure $\mu $…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…

Consider a random walk $(S_n:n\geq0)$ with drift $-\mu$ and $S_0=0$. Assuming that the increments have exponential moments, negative mean, and are strongly nonlattice, we provide a complete asymptotic expansion (in powers of $\mu>0$) that…

We establish a new comparison between the Legendre transform of the cumulant generating function and the half-space depth of an arbitrary log-concave probability distribution on the real line, that carries on to the multidimensional…

This paper discovers a new phenomenon about the Duffin-Schaeffer conjecture, which claims that $\lambda(\cap_{m=1}^{\infty}\cup_{n=m}^{\infty}{\mathcal E}_n)=1$ if and only if $\sum_n\lambda({\mathcal E}_n)=\infty$, where $\lambda$ denotes…

This short note gives a sufficient condition for having the class of polynomials dense in the space of square integrable functions with respect to a finite measure dominated by the Lebesgue measure in the real line, here denoted by $L^2$.…

Given a probability measure $\mu $ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $\varphi_{\mu }(x)=\inf\{\mu (H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of…

We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure $\lambda$ and an invariant random measure $\mu$ of unit intensity to be finite. We show that for \emph{any} such random measure the…

We study the dimensional Brunn-Minkowski inequality for even log-concave probability measures $\mu$ on $\mathbb{R}^n$ via an analytic approach based on diffusion operators and gradient estimates. Our main result asserts that for every pair…

Given all (finite) moments of two measures $\mu$ and $\lambda$ on $\R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $\mu=\nu+\psi$ with $\nu\ll\lambda$ and $\psi\perp\lambda$. When$\nu$ has a density in…

Let $\mu$ be the logarithmic equilibrium measure on a compact set $\gamma \subset \mathbb{R}^{d}$. We prove that $\mu$ is absolutely continuous with respect to the length measure on the part of $\gamma$ which can be locally expressed as the…

Given $d\ge 1$, we provide a construction of the random measure - the critical Gaussian Multiplicative Chaos - formally defined $e^{\sqrt{2d}X}\mathrm{d} \mu$ where $X$ is a $\log$-correlated Gaussian field and $\mu$ is a locally finite…

Let $p\geq 1$, $\eps >0$, $r\geq (1+\eps) p$, and $X$ be a $(-1/r)$-concave random vector in $\R^n$ with Euclidean norm $|X|$. We prove that $(\E |X|^{p})^{1/{p}}\leq c (C(\eps) \E|X|+\sigma_{p}(X))$, where $\sigma_{p}(X)=\sup_{|z|\leq…

In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there…

We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincar\'e constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations…

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent unbiased coin tosses. We prove that each…

Let $G$ be a locally compact group and $\mu$ be a probability measure on $G$. We consider the convolution operator $\lambda_1(\mu)\colon L_1(G)\to L_1(G)$ given by $\lambda_1(\mu)f=\mu \ast f$ and its restriction $\lambda_1^0(\mu)$ to the…

Let $\mu$ be an even Borel probability measure on ${\mathbb R}$. For every $N>n$ consider $N$ independent random vectors $\vec{X}_1,\ldots ,\vec{X}_N$ in ${\mathbb R}^n$, with independent coordinates having distribution $\mu $. We establish…