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This article is cited in 23 scientific papers (total in 23 papers)
Short Communications
The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero
R. Ibragimova, Sh. Sharahmetovb a Central Michigan University
b Tashkent State University
Abstract:
Let $\xi_1, \ldots, \xi_n$ be independent random variables with $\mathbf{E}\xi_i=0,$ $\mathbf{E}|\xi_i|^t<\infty$, $t>2$, $i=1,\ldots, n,$ 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 $$ \mathbf{E}|S_n|^t\le C(t) \max \Bigg(\sum_{i=1}^n\mathbf{E}|\xi_i|^t,\ \Bigg(\sum_{i=1}^n \mathbf{E}\xi_i^2\Bigg)^{t/2}\Bigg) $$ for $t=2m,$ $m\in \mathbf{N},$ 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 $$ \overline C(2m)=\mathbf{E}(\theta-1)^{2m}, $$ where $\theta $ is a Poisson random variable with parameter 1.
Keywords:
Rosenthal inequality, zero mean random variables, moment, Poisson random variable.
Received: 30.03.1998 Revised: 15.03.1999
Citation:
R. Ibragimov, Sh. Sharahmetov, “The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero”, Teor. Veroyatnost. i Primenen., 46:1 (2001), 134–138; Theory Probab. Appl., 46:1 (2002), 127–132
Linking options:
https://www.mathnet.ru/eng/tvp4011https://doi.org/10.4213/tvp4011 https://www.mathnet.ru/eng/tvp/v46/i1/p134
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