关于Rademacher-Gaussian比较的多维球不变扩张
On a multidimensional spherically invariant extension of the Rademacher--Gaussian comparison
摘要：
本文证明了对所有实$u$，有\begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*}，其中$U_1,\dots,U_n$是$\mathbb{R}^d$中单位球面上均匀分布的独立随机向量，$a_1,\dots,a_n$是任意实数，$a:=\sqrt{(a_1^2+\dots+a_n^2)/d}$，$Z_d$是$\mathbb{R}^d$中的标准正态随机向量，$c=2e^3/9=4.46\dots$ .这个常数因子大约是Nayar和Tkocz最近结果的$89$倍，他们用不同的方法证明了Oleszkiewicz的相应猜想。作为直接应用，给出了任意独立球不变随机向量和范数的尾概率的一个相应上界。
英文摘要：
It is shown that \begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*} for all real $u$, where $U_1,\dots,U_n$ are independent random vectors uniformly distributed on the unit sphere in $\mathbb{R}^d$, $a_1,\dots,a_n$ are any real numbers, $a:=\sqrt{(a_1^2+\dots+a_n^2)/d}$, $Z_d$ is a standard normal random vector in $\mathbb{R}^d$, and $c=2e^3/9=4.46\dots$. This constant factor is about $89$ times as small as the one in a recent result by Nayar and Tkocz, who proved, by a different method, a corresponding conjecture by Oleszkiewicz. As an immediate application, a corresponding upper bound on the tail probabilities for the norm of the sum of arbitrary independent spherically invariant random vectors is given.
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