Document Type

Article

Publication Date

9-19-2016

Abstract

It is shown that P(ǁa1U1+ · · · +anUnǁ > u) ≤ cP(aǁZdǁ> u) for all real u, where U1, : : : , Un are independent random vectors uniformly distributed on the unit sphere in Rd, a1, : : : , an are any real numbers, a := √(a21 + · · · · + a2n ) ∕ d, Zd is a standard normal random vector in Rd, and c = 2e3 ∕ 9 = 4.46 . . . .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.

Publisher's Statement

© 2016 Project Euclid. Deposited in compliance with publisher policies. Publisher's version of record: http://dx.doi.org/10.1214/16-ECP23

Publication Title

Electronic Communications in Probability

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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Mathematics Commons

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