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.
Highlights
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
It is interesting to note that, in the case l=0, the expression on the right-hand side of (2), with the inner sum taken over all natural m1>m2>...>mr and j1,...,jr satisfying the conditions m1j1+...+mrjr=2m, j1+...+jr =j, equals to the best constant the analogue of inequality (1) for nonnegative r.v.’s
Where the inner sum is taken over all natural m1>m2>...>mr and j1,..., jr, satisfying the conditions m1j1+...+mrjr=2m, j1+...+jr=j, m i ≠ 2s −1, i = 1,2,...,r , s = 1,2,...,l
Summary
OPTIMAL CONSTANTS IN THE ROSENTHAL INEQUALITY FOR RANDOM VARIABLES WITH ZERO ODD MOMENTS1. Marat Ibragimov Department of Probability Theory, Tashkent State Economics University
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