Geodesic
On an inequality and on the related characterization of the normal distribution
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 209-218 
A. A. Borovkov; S. A. Utev. On an inequality and on the related characterization of the normal distribution. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 209-218. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a0/
@article{TVP_1983_28_2_a0,
     author = {A. A. Borovkov and S. A. Utev},
     title = {On an inequality and on the related characterization of the normal distribution},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {209--218},
     year = {1983},
     volume = {28},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a0/}
}
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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.