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

By Rustam Ibragimov, Sh. Sharakhmetov. 2001

57 citations recorded by the source corpus.

Let $\xi_1, \ldots, \xi_n$ be independent random variables with ${\bf E}\xi_i=0,$ ${\bf E}|\xi_i|^t 2$, $i=1,\ldots, n,$ 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.

Subfields: Applied Mathematics, Mathematical Physics

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In Applied Mathematics

1. Weidong Liu
2. Han Xiao
3. Max Fathi
4. Rustam Ibragimov
5. Stephen Tu
6. Wei Biao Wu
7. B. Recht
8. Yuta Koike

In Mathematical Physics

1. Wei Biao Wu
2. Weidong Liu
3. Max Fathi
4. Han Xiao
5. Rustam Ibragimov
6. Stephen Tu
7. B. Recht
8. Xiaofeng Shao

Citations in the corpus

• Probability and moment inequalities under dependence (2013, background): “(1.4) The calculation of the constant Bp has been extensively discussed in the literature; see Pinelis and Utev (1984), Johnson, Schechtman, and Zinn (1985), Ibragimov and Sharakhmetov (2002), among others.”
• Safely Learning to Control the Constrained Linear Quadratic Regulator (2018, background): “By Rosenthal’s inequality, we have that for an absolute constant–we can take C = 4 (see e.g. [29]).”
• Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series (2006, background): “The importance of the generalized Eyraud–Farlie–Gumbel–Morgenstern copulas and cdf’s stems, in particular, from the fact that, as shown in Sharakhmetov and Ibragimov [70], they completely characterize joint distributions of two-valued r.v.’s. Taking n = 2, g̃1,2(t1, t2) = θc(t1, t2), where c is a co”
• Stein kernels and moment maps (2018, background): “, xn), and Rosenthal’s inequality for sums of independent random variables [16] yields”