On an inequality and on the related characterization of the normal distribution

A. A. Borovkov   ; S. A. Utev

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 209-218

Résumé

We obtain the conditions on the distribution of the random variable $\xi$ under which the inequality $$ \mathbf Dg(\xi)\le c\mathbf E(g'(\xi))^2 $$ holds for any differentiable function $g$. Some properties of the functional $$ U_\xi=\sup_g\frac{\mathbf Dg(\xi)}{\mathbf D\xi\mathbf E(g'(\xi))^2} $$ are investigated also. It is proved that $U_\xi\ge 1$ and that $U_\xi=1$ iff the random variable $\xi$ has the normal distribution. The theorem of continuity is true as well: if $U_{\xi_n}\to 1$ as $n\to\infty$, then the distributions of $\xi_n^{(1)}=(\xi_n-\mathbf E\xi_n)/\sqrt{D\xi_n}$ converge to the normal one.