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}.

Перевод пока недоступен