Abstract

We obtain estimates for the best constant in the Rosenthal inequality E | ∑ i = 1 n ξ i | 2 m ⩽ C ( 2 m ) max ( ∑ i = 1 n E ξ i 2 m , ( ∑ i = 1 n E ξ i 2 ) m ) for independent random variables ξ 1 , … , ξ n with l zero first odd moments, l ⩾ 1 . The estimates are sharp in the extremal cases l = 1 and l = m , that is, in the cases of random variables with zero mean and random variables with m zero first odd moments.

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