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

Abstract

It is shown that \[ \mathsf{P} (\|a_1U_1+\dots +a_nU_n\|>u)\le c\,\mathsf{P} (a\|Z_d\|>u) \] 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.