On a multidimensional spherically invariant extension of the Rademacher--Gaussian comparison

TL;DR

This paper extends the Rademacher--Gaussian comparison to multidimensional spherically invariant distributions, providing a tighter probability bound with a smaller constant factor, and applies it to sums of independent spherically invariant vectors.

Contribution

It introduces a new inequality for spherically invariant vectors, improving the constant factor compared to previous results, and applies it to bound tail probabilities of sums.

Findings

Derived a probability inequality with a smaller constant factor (about 4.46)

Provided an upper bound on tail probabilities for sums of spherically invariant vectors

Improved upon recent results by Nayar and Tkocz

Abstract

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.