Related papers: Variance bounds in product measures without expone…

We discuss optimal constants in a recent result of Rudelson and Vershynin on marginal densities. We show that if $f$ is a probability density on $\R^n$ of the form $f(x)=\prod_{i=1}^n f_i(x_i)$, where each $f_i$ is a density on $\R$, say…

Given $\epsilon \in (0,1)$, a probability measure $\mu$ on $\Omega\subset\mathbb{R}^p$ and a semi-algebraic set $K\subset X\times\Omega$, we consider the feasible set $X^*_\epsilon=\{x\in X:{\rm Prob}[(x,\omega)\in K]\geq 1-\epsilon\}$…

We introduce a new measure of robustness for statistical estimators, which we call \emph{empirical sensitivity}. An estimator $\hat \theta$ has bounded empirical sensitivity if, with high probability over a dataset $X = (X_1, \dots, X_n)…

We consider a Markov chain on $\mathbb{R}^d$ with invariant measure $\mu$. We are interested in the rate of convergence of the empirical measures towards the invariant measure with respect to various dual distances, including in particular…

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…

Let $\mu$ be a probability measure on $\mathbb{R}$. We give conditions on the Fourier transform of its density for functionals of the form $H(a)=\int_{\mathbb{R}^n}h(\langle a,x\rangle)\mu^n(dx)$ to be Schur monotone. As applications, we…

New sum and product uncertainty relations, containing variances of three or four observables, but not containing explicitly their covariances, are derived. One of consequences is the new inequality, giving a nonzero lower bound for the…

In this manuscript, we consider the Langevin dynamics on $\mathbb{R}^d$ with an overdamped vector field and driven by multiplicative Brownian noise of small amplitude $\sqrt{\epsilon}$, $\epsilon>0$. Under suitable assumptions on the vector…

Outer measures can be used for statistical inference in place of probability measures to bring flexibility in terms of model specification. The corresponding statistical procedures such as Bayesian inference, estimators or hypothesis…

We introduce an augmented form of the van Trees inequality, that yields uniformly tighter lower bounds on the minimax squared Bayes risk of estimators compared with the classical van Trees inequality. Our augmented inequality also…

Let $\mu$ be a finite Radon measure on an open set $\Omega\subset\mathbb{R}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian…

Many management decisions involve accumulated random realizations for which only the first and second moments of their distribution are available. The sharp Chebyshev-type bound for the tail probability and Scarf bound for the expected loss…

Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near…

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses…

Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of…

We show that if $(X, \mu, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ is a countable partition of $(X, \mu)$, then the limit $$ \lim_{n, k \to \infty} \mathbb{E} \left[ \frac{1}{k} \sum_{j = 0}^{k - 1} f…

We prove that for a minimal rotation T on a 2-step nilmanifold and any measure mu, the push-forward T^n(mu) of mu under T^n tends toward Haar measure if and only if mu projects to Haar measure on the maximal torus factor. For an arbitrary…

One of the goals of this article is to define a an unified setting adapted to the description of means (normalized integrals or invariant means) on an infinite product of measured spaces with infinite measure. We first remark that some…

For each $n$, let $A_n=(\sigma_{ij})$ be an $n\times n$ deterministic matrix and let $X_n=(X_{ij})$ be an $n\times n$ random matrix with i.i.d. centered entries of unit variance. We study the asymptotic behavior of the empirical spectral…

We observe that approximate copies of the function $\Lambda _{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} \Lambda _{n}(x)=\exp \left( -x_{1}-\pi \sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails…