Electronic Communications in Probability

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

Iosif Pinelis

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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

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

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

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Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G15: Gaussian processes 60G50: Sums of independent random variables; random walks

probability inequalities generalized moment comparison tail comparison sums of independent random vectors Gaussian random vectors uniform distribution on spheres

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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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