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2016 On a multidimensional spherically invariant extension of the Rademacher–Gaussian comparison
Iosif Pinelis
Electron. Commun. Probab. 21: 1-5 (2016). DOI: 10.1214/16-ECP23


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.


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Iosif Pinelis. "On a multidimensional spherically invariant extension of the Rademacher–Gaussian comparison." Electron. Commun. Probab. 21 1 - 5, 2016.


Received: 7 April 2016; Accepted: 7 September 2016; Published: 2016
First available in Project Euclid: 19 September 2016

zbMATH: 1353.60045
MathSciNet: MR3564214
Digital Object Identifier: 10.1214/16-ECP23

Primary: 60E15
Secondary: 60G15 , 60G50

Keywords: Gaussian random vectors , generalized moment comparison , Probability inequalities , sums of independent random vectors , tail comparison , uniform distribution on spheres


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