# The exact constant in the Rosenthal inequality for random variables with mean zero

 Title: The exact constant in the Rosenthal inequality for random variables with mean zero Author: Ibragimov, Rustam; Sharakhmetov, Shaturgun Note: Order does not necessarily reflect citation order of authors. Citation: Ibragimov, Rustam and Shaturgun Sharakhmetov. 2002. The exact constant in the Rosenthal inequality for random variables with mean zero. Theory of Probability and Its Applications 46(1): 127-132. Full Text & Related Files: ibragimov_meanzero.pdf (114.9Kb; PDF) Abstract: Let $\xi_1, \ldots, \xi_n$ be independent random variables with ${\bf E}\xi_i=0,$ ${\bf E}|\xi_i|^t<\infty$, $t>2$, $i=1,\ldots, n,$ and let $S_n=\sum_{i=1}^n \xi_i.$ In the present paper we prove that the exact constant ${\overline C}(2m)$ in the Rosenthal inequality $${\bf E}|S_n|^t\le C(t) \max \Bigg(\sum_{i=1}^n{\bf E}|\xi_i|^t,\ \Bigg(\sum_{i=1}^n {\bf E}\xi_i^2\Bigg)^{t/2}\Bigg)$$ for $t=2m,$ $m\in {\bf N},$ is given by $$\overline C(2m)=(2m)! \sum_{j=1}^{2m} \sum_{r=1}^j \sum \prod_{k=1}^r \frac {(m_k!)^{-j_k}} {j_k!},$$ where the inner sum is taken over all natural $m_1 > m_2 > \cdots > m_r > 1$ and $j_1, \ldots, j_r$ satisfying the conditions $m_1j_1+\cdots+m_rj_r=2m$ and $j_1+\cdots+j_r=j$. Moreover $$\overline C(2m)={\bf E}(\theta-1)^{2m},$$ where $\theta$ is a Poisson random variable with parameter 1. Published Version: http://dx.doi.org/10.1137/S0040585X97978762 Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:2623703

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