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Teoriya Veroyatnostei i ee Primeneniya, 2001, Volume 46, Issue 1, Pages 134–138
DOI: https://doi.org/10.4213/tvp4011
(Mi tvp4011)
 

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
English version:
Theory of Probability and its Applications, 2002, Volume 46, Issue 1, Pages 127–132
DOI: https://doi.org/10.1137/S0040585X97978762
Bibliographic databases:
Document Type: Article
Language: Russian
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
Citation in format AMSBIB
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\by R.~Ibragimov, Sh.~Sharahmetov
\paper The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero
\jour Teor. Veroyatnost. i Primenen.
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\issue 1
\pages 134--138
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\crossref{https://doi.org/10.4213/tvp4011}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1968709}
\zmath{https://zbmath.org/?q=an:1008.60038}
\transl
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 1
\pages 127--132
\crossref{https://doi.org/10.1137/S0040585X97978762}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000174464700009}
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  • This publication is cited in the following 23 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
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