Related papers: A uniform Dvoretzky-Kiefer-Wolfowitz inequality

Let $\Omega\subset{\mathbb R}^2$ be a bounded domain on which Hardy's inequality holds. We prove that $[\exp(u^2)-1]/\delta^2\in L^1(\Omega)$ if $u\in H^1_0(\Omega)$, where $\delta$ denotes the distance to $\partial\Omega$. The…

A probability distribution over {-1, 1}^n is (eps, k)-wise uniform if, roughly, it is eps-close to the uniform distribution when restricted to any k coordinates. We consider the problem of how far an (eps, k)-wise uniform distribution can…

Let $\mathbf{X} = (X_i)_{1\leq i \leq n}$ be an i.i.d. sample of square-integrable variables in $\mathbb{R}^d$, \GB{with common expectation $\mu$ and covariance matrix $\Sigma$, both unknown.} We consider the problem of testing if $\mu$ is…

Given a Lipschitz function $f:\{1,...,d\}^\mathbb{N} \to \mathbb{R}$, for each $\beta>0$ we denote by $\mu_\beta$ the equilibrium measure of $\beta f$ and by $h_\beta$ the main eigenfunction of the Ruelle Operator $L_{\beta f}$. Assuming…

Let $(\varphi_i)_{i=1}^n$ be mutually orthogonal functions on a probability space such that $\|\varphi_i\|_\infty \leq 1 $ for all $i \in [n]$. Let $\alpha > 0$. Let $\Phi(u) = u^2 \log^{\alpha}(u)$ for $u \geq u_{0}$, and $\Phi(u) =…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

This paper studies estimation of and inference on a distribution function $F$ that is concave on the nonnegative half line and admits a density function $f$ with potentially unbounded support. When $F$ is strictly concave, we show that the…

For $0 \leq \beta < \alpha < 1$ the distribution $\mathcal{H}$ over Boolean functions $h \colon \{-1, 1\}^d \to \{-1, 1\}$ that minimizes the expression \begin{equation*} \rho_{\alpha, \beta} = \frac{\log(1/\Pr_{\substack{h \sim \mathcal{H}…

For a uniform process $\{ X_t: t\in E\}$ (by which $X_t $ is uniformly distributed on $(0,1)$ for $t\in E$) and a function $w(x)>0$ on $(0,1)$, we give a sufficient condition for the weak convergence of the empirical process based on $\{…

On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\beta \in [0,1)$ and $w_0(x) = |\log |x||^\beta $ then $$ \sup_{\int_B…

In the present paper, we show that, for every $\delta>0$, the function $(\log {\mathcal{L}}(s))^{(m)}$, where $m\in {\mathbb{N}} \cup \{ 0\}$ and ${\mathcal{L}} (s) := \sum_{n=1}^\infty a(n) n^{-s}$ is an element of the Selberg class…

Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\le\|f\|_\infty$. We give a new and…

Let $d\in\mathbb N$ and $f$ be a function in the Orlicz class $L(\log^+L)^{d-1}$ defined on the unit cube $[0,1]^d$ in $\mathbb{R}^d$. Given partitions $\Delta_1,\ldots,$ $\Delta_d$ of $[0,1]$, we first prove that the orthogonal projection…

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high…

Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:= \{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $\alpha \geq 0$, we consider the following class $$\mathcal{W}^0_{\mathcal{H}}(\alpha):=…

Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,...,f_n:{-1,1}^n\to X$ satisfy \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n…

We give a new proof of the Gromov theorem: For any $C>0$ and integer $n>1$ there exists a function $\Delta_{C,n}$ such that if the Gromov--Hausdorff distance between complete Riemannian $n$-manifolds $V$ and $W$ is not greater than…

We obtain exact conditions guaranteeing that any global weak solution of the differential inequality $$ \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, u) \ge g (|u|) \quad \mbox{in } {\mathbb R}^n $$ is trivial, where $m, n \ge 1$ are…

We prove that, in the space of all probabilistic continuous functions from a probabilistic metric space G to the set $\Delta$ + of all cumulative distribution functions vanishing at 0, the space of all 1-Lipschitz functions is compact if…

For an analytic and univalent function $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ with the normalization $f(0)=0=f'(0)-1$, the logarithmic coefficients $\gamma_n$ are defined by $\log \frac{f(z)}{z}= 2\sum_{n=1}^{\infty}…