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, $1

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

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