The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${mathbb R}^n$. We provide upper bounds for the multi-integral expression begin{equation*}|{bf t}|_{C^s,K}=int_{C}cdotsint_{C}Big|sum_{j=1}^st_jx_jBig|_K,dx_1cdots dx_send{equation
*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to randomized vector balancing problems.
In this paper we study bounds for the total variation distance between two second degree polynomials in normal random variables provided that they essentially depend on at least three variables.
Avikainen showed that, for any $p,q in [1,infty)$, and any function $f$ of bounded variation in $mathbb{R}$, it holds that $mathbb{E}[|f(X)-f(widehat{X})|^{q}] leq C(p,q) mathbb{E}[|X-widehat{X}|^{p}]^{frac{1}{p+1}}$, where $X$ is a one-dimensional r
andom variable with a bounded density, and $widehat{X}$ is an arbitrary random variable. In this article, we will provide multi-dimensiona
The TCP window size process appears in the modeling of the famous Transmission Control Protocol used for data transmission over the Internet. This continuous time Markov process takes its values in [0, infty), is ergodic and irreversible. The sample
paths are piecewise linear deterministic and the whole randomness of the dynamics comes from the jump mechanism. The aim of the present paper is to provide quantitative estimates for the exponential convergence to equilibrium, in terms of the total variation and Wasserstein distances.
In this paper, we study the asymptotic thin-shell width concentration for random vectors uniformly distributed in Orlicz balls. We provide both asymptotic upper and lower bounds on the probability of such a random vector $X_n$ being in a thin shell o
f radius $sqrt{n}$ times the asymptotic value of $n^{-1/2}left(mathbb Eleft[| X_n|_2^2right]right)^{1/2}$ (as $ntoinfty$), showing that in certain ranges our estimates are optimal. In particular, our estimates significantly improve upon the currently best known general Lee-Vempala bound when the deviation parameter $t=t_n$ goes down to zero as the dimension $n$ of the ambient space increases. We shall also determine in this work the precise asymptotic value of the isotropic constant for Orlicz balls. Our approach is based on moderate deviation principles and a connection between the uniform distribution on Orlicz balls and Gibbs measures at certain critical inverse temperatures with potentials given by Orlicz functions, an idea recently presented by Kabluchko and Prochno in [The maximum entropy principle and volumetric properties of Orlicz balls, J. Math. Anal. Appl. {bf 495}(1) 2021, 1--19].