Related papers: $L^1$ bounds in normal approximation

In this paper we discuss the global symmetries and the renormalizibility of Lee-Wick scalar QED. In particular, in the "auxiliary-field" formalism we identify softly broken SO(1,1) global symmetries of the theory. We introduce SO(1,1)…

We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a…

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…

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration…

We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite…

Likelihood ratio tests and the Wilks theorems have been pivotal in statistics but have rarely been explored in network models with an increasing dimension. We are concerned here with likelihood ratio tests in the $\beta$-model for…

It is well known that the entropy $H(X)$ of a discrete random variable $X$ is always greater than or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tight…

Motivated by the simulation of stable random fields, we consider the issue of discrete approximations of independently scattered stable noise. Two approaches are proposed: grid approximations available when the underlying space is $\bbR^d$…

For an ergodic Brownian diffusion with invariant measure $\nu$, we consider a sequence of empirical distributions ($\nu$n) n$\ge$1 associated with an approximation scheme with decreasing time step ($\gamma$n) n$\ge$1 along an adapted…

The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach…

We exhibit a range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $d\geq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds,…

We prove sharp homogeneous improvements to $L^1$ weighted Hardy inequalities involving distance from the boundary. In the case of a smooth domain, we obtain lower and upper estimates for the best constant of the remainder term. These…

In this set of four lectures I will discuss some aspects of the Standard Model (SM) as a quantum field theory and related phenomenological observations which have played a crucial role in establishing the $SU(2)_{L} \times U(1)_{Y}$ gauge…

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $\varphi :\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^…

Let X be a real or complex Hilbert space of finite but large dimension d, let S(X) denote the unit sphere of X, and let u denote the normalized uniform measure on S(X). For a finite subset B of S(X), we may test whether it is approximately…

In this paper, we proposed a new lifetime distribution namely generalized weighted Lindley (GLW) distribution. The GLW distribution is a useful generalization of the weighted Lindley distribution, which accommodates increasing, decreasing,…

The $L^k$-Wasserstein distance $\mathbb{W}_k (k\ge 1)$ and the probability distance $\mathbb{W}_\psi$ induced by a concave function $\psi$, are estimated between different diffusion processes with singular coefficients. As applications, the…

Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from…

Suppose $\alpha, \beta$ are Lipschitz strongly concave functions from $[0, 1]$ to $\mathbb{R}$ and $\gamma$ is a concave function from $[0, 1]$ to $\mathbb{R}$, such that $\alpha(0) = \gamma(0) = 0$, and $\alpha(1) = \beta(0) = 0$ and…

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of…