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

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.