## Electronic Communications in Probability

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

Iosif Pinelis

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 67, 5 pp.

Dates
Accepted: 7 September 2016
First available in Project Euclid: 19 September 2016

https://projecteuclid.org/euclid.ecp/1474301119

Digital Object Identifier
doi:10.1214/16-ECP23

Mathematical Reviews number (MathSciNet)
MR3564214

Zentralblatt MATH identifier
1353.60045

#### Citation

Pinelis, Iosif. On a multidimensional spherically invariant extension of the Rademacher–Gaussian comparison. Electron. Commun. Probab. 21 (2016), paper no. 67, 5 pp. doi:10.1214/16-ECP23. https://projecteuclid.org/euclid.ecp/1474301119

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