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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 1, Pages 134–138 (Mi tvp4011)  

This article is cited in 12 scientific papers (total in 12 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
$$ \mathbf{E}|S_n|^t\le C(t) \max (\sum_{i=1}^n\mathbf{E}|\xi_i|^t, (\sum_{i=1}^n \mathbf{E}\xi_i^2)^{t/2}) $$
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.


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English version:
Theory of Probability and its Applications, 2002, 46:1, 127–132

Bibliographic databases:

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

Citation in format AMSBIB
\by R.~Ibragimov, Sh.~Sharahmetov
\paper The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero
\jour Teor. Veroyatnost. i Primenen.
\yr 2001
\vol 46
\issue 1
\pages 134--138
\jour Theory Probab. Appl.
\yr 2002
\vol 46
\issue 1
\pages 127--132

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    This publication is cited in the following articles:
    1. Ibragimov R., Sharakhmetov S., “Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal inequalities for symmetric statistics”, Scand. J. Statist., 26:4 (1999), 621–633  crossref  mathscinet  zmath  isi  scopus
    2. Nze P.A., Doukhan P., “Weak dependence: models and applications to econometrics”, Econometric Theory, 20:6 (2004), 995–1045  crossref  mathscinet  zmath  isi  scopus
    3. S. V. Nagaev, “On probability and moment inequalities for supermartingales and martingales”, Theory Probab. Appl., 51:2 (2007), 367–377  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. E. L. Presman, “Estimation of the Constant in a Burkholder Inequality for Supermartingales and Martingales”, Theory Probab. Appl., 53:1 (2009), 173–179  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. Ibragimov M., Ibragimov R., “Optimal constants in the Rosenthal inequality for random variables with zero odd moments”, Statist. Probab. Lett., 78:2 (2008), 186–189  crossref  mathscinet  zmath  isi  scopus
    6. Abadie A., Diamond A., Hainmueller J., “Synthetic control methods for comparative case studies: estimating the effect of California's tobacco control program”, J. Amer. Statist. Assoc., 105:490 (2010), 493–505  crossref  mathscinet  zmath  isi  scopus
    7. Hansen B.E., “The Integrated Mean Squared Error of Series Regression and a Rosenthal Hilbert-Space Inequality”, Economet. Theory, 31:2 (2015), 337–361  crossref  mathscinet  zmath  isi  scopus
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    9. Cekanavicius V., “Approximation Methods in Probability Theory”, Approximation Methods in Probability Theory, Universitext, Springer International Publishing Ag, 2016, 1–274  crossref  mathscinet  isi
    10. Cadre B., Klutchnikoff N., Massiot G., “Minimax Regression Estimation For Poisson Coprocess”, ESAIM-Prob. Stat., 21 (2017), 138–158  crossref  mathscinet  isi  scopus
    11. Fathi M., “Stein Kernels and Moment Maps”, Ann. Probab., 47:4 (2019), 2172–2185  crossref  isi
    12. Sumritnorrapong P., Neammanee K., Suntornchost J., “An Improvement of a Non-Uniform Bound For Combinatorial Central Limit Theorem”, Commun. Stat.-Theory Methods, 48:9 (2019), 2129–2146  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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