Sankhya: The Indian Journal of Statistics

2002, Volume 64, Series A, Pt. 1, pp. 42--56

ON EXTREMAL PROBLEMS AND BEST CONSTANTS IN MOMENT INEQUALITIES

By

RUSTAM IBRAGIMOV, Yale University

and

SHATURGUN SHARAKHMETOV, Tashkent State Economics University

SUMMARY. In the present paper, we show that the best constant A*(t,g) in the Rosenthal-type inequality with an arbitrary balancing factor g>0
$$E( splaystyle\mathop\Sigma^n_{i=1} X_i \right)^t \le A(t, \gamma) \max \left( \gamma \displaystyle\mathop\Sigma^n_{i=1} EX^t_i, \left( \displaystyle\mathop\Sigma^n_{i=1} X_i \right)^t \right)$$ \vskip10pt \noindent for independent nonnegative random variables $X_1, \ldots, X_n$ with finite $t$-th moment, $12,$$ where $Z(\gamma^{1/(t-1)})$ is a Poisson random variable with parameter $\gamma^{1/(t-1)}.$ In addition to that, we obtain estimates for the best constants in analogues of the above inequality for independent random variables with a set of zero odd moments that generalize and complement the results known for mean-zero random variables and symmetric random variables.

AMS (1991) subject classification. Primary 60E15, 60F25, 60G50.

Key words and phrases. Sums of independent random variables, moment inequalities, Rosenthal's inequalities, best constants.

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