Rosenthal Inequality Research Articles

Syndrome de Melkersson-Rosenthal et déficit en C1 inhibiteur fonctionnel

In the present paper, we obtain the explicit expression for the best constant in the Rosenthal inequality for nonnegative random variables.

On the Rosenthal's inequality for locally square integrable martingales

Moment inequalities for locally square integrable martingales are considered. The growth rates of the constants in Rosenthal's inequality for locally square integrable martingales and Burkholder–Gundy inequality for martingales with symmetric jumps are given.

The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero

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 ...

The best constant in the Rosenthal inequality for nonnegative random variables

In the present paper, we obtain the explicit expression for the best constant in the Rosenthal inequality for nonnegative random variables.

Exact Estimates for Moments of Random Bilinear Forms

The present paper concentrates on the analogues of Rosenthal's inequalities for ordinary and decoupled bilinear forms in symmetric random variables. More specifically, we prove the exact moment inequalities for these objects in terms of moments of their individual components. As a corollary of these results we obtain the explicit expressions for the best constant in the analogues of Rosenthal's inequality for ordinary and decoupled bilinear forms in identically distributed symmetric random variables in the case of the fixed number of random variables.

On the rosenthal inequality for mixing fields

A proof of the Rosenthal inequality for α-mixing random fields is given. The statements and proofs are modifications of the corresponding results obtained by Doukhan and Utev.

Rosenthal's inequality for point process martingales

Moment inequalities for point process martingales are considered. Our main result is a point process analogue of Rosenthal's inequality for discrete-time martingales. It is also noted that this inequality generalises in a simple way to marked point process martingales.

Rosenthal's inequality for LPQD sequences

We note that Rosenthal's type inequalities for LPQD random variables (r.v.'s) are written along the line of Birkel [Birkel, T., 1988. Moment bounds for associated sequences. Ann. Probab. 16, 1184–1193]. A generalization of this result is obtained.

Some Refinements of Probabilistic andMoment Inequalities

The probabilistic inequality established for sums of independent random variables refines one of the estimates given in the paper by D. H. Fuk and S. V. Nagaev, [Theory Probab. Appl., 16 (1971), pp. 660--675. This inequality entails, in particular, the recent results of I. Pinelis, [Ann. Probab., 7 (1979), pp. 745--789, on estimation of constants in the Rosenthal inequality.

Short Communications: On an Exact Constant for the Rosenthal Inequality

Let $\xi_1\lz \xi_n$ be independent random variables having symmetric distribution with finite pth moment, $2 < p < \ iy$. It is shown that the precise constant $C_p^*$ in Rosenthal's inequality $$ \bigg\|\sion\xi_i\bigg\|_p\l C_p\max \bigg(\bigg\|\sion \xi_i\bigg\|_2,\ \bigg(\sion \|\xi_i\|_p^p\bigg)^{1/p}\bigg) $$ has the form $$ \eqalignno{ C_p^*&=\bigg(1+{2^{p/2}\over \pi^{1/2}}\,\Gm\Big({p+1\over 2}\Big)\bigg)^{1/p},\q 2 < p < 4, \cr C_p^*&=\|\xi_1-\xi_2\|_p,\q p\g 4, \cr } $$ where $\Gm(\ap)=\iny x^{\ap-1}e^{-x}dx$ and $\xi_1,\xi_2$ are independent Poisson random variables with parameter~0.5. It is proved also that $$ \lim_{p\to\iy} C_p^*{\log p\over p}={1\over e}.

Rosenthal's inequality and the characterization of Lp spaces

Rosenthal's inequality and the characterization of Lp spaces

On Rosenthal's inequality and rearrangement invariant spaces

On Rosenthal's inequality and rearrangement invariant spaces

The rosenthal inequality in symmetric spaces

The rosenthal inequality in symmetric spaces

Best Constants in Martingale Version of Rosenthal's Inequality

The following generalization of Rosenthal's inequality was proved by Burkholder: $A^{-1}_p\{\|s(f)\|_p + \|d^\ast\|_p\} \leq \|f^\ast\|_p \leq B_p\{\|s(f)\|_p + \|d^\ast\|_p\},$ for all martingales $(f_n)$. It is known that $A_p$ grows like $\sqrt{p}$ as $p \rightarrow \infty$. In this paper we prove that the growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\ln p$.