论文标题
通过中等偏差和吉布斯测量
Thin-shell concentration for random vectors in Orlicz balls via moderate deviations and Gibbs measures
论文作者
论文摘要
在本文中，我们研究了均匀分布在Orlicz球中的随机向量的渐近薄壳宽度浓度。我们以这种随机向量$ x_n $的概率在半径$ \ sqrt {n} $乘以$ n^{ - 1/2} \ left（\ mathb e \ weft [\ | x_n \ | _ | _2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2^2， $ n \ to \ infty $），表明在某些范围内，我们的估计值是最佳的。特别是，当偏差参数$ t = t_n $随着环境空间的尺寸$ n $的增加时，我们的估计值大大改善了当前最著名的一般Lee-vempala绑定的限制。在这项工作中，我们还应确定Orlicz球的各向同性常数的精确渐近值。我们的方法基于中等偏差原理，以及在某些关键的逆温度下的Orlicz球上的均匀分布与Orlicz函数给出的电势，这是Kabluchko和prochno在[Orlicz Balls的最大熵原理和最大熵原理和体积特性，J。Maths中的最大熵原理和体积特性。肛门。应用。 {\ bf 495}（1）2021，1--19]。
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 of radius $\sqrt{n}$ times the asymptotic value of $n^{-1/2}\left(\mathbb E\left[\| X_n\|_2^2\right]\right)^{1/2}$ (as $n\to\infty$), 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].
