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.
Electronic Communications in Probability
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Pinelis, Iosif, "On a multidimensional spherically invariant extension of the Rademacher-Gaussian comparison" (2016). Department of Mathematical Sciences Publications. 4.