A~generalization of theorems due to H.~Cramer and Yu.\,V.~Linnik--V.\,P.~Skitovi\v c

G. P. Chistyakov

Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 2, pp. 345-350

Résumé

Let $B$ be a class of functions $V(x)$ with bounded variation on $(-\infty,\infty)$ satisfying the conditions: 1) $\int_{-\infty}^\infty dV(x)=1$; 2) $V(x)=\omega_1(x)-\omega_2(x)$; where $\omega_j(x)$ are nondecreasing functions $\omega_j(x)+\omega_j(-x)=2\omega_j(0)$, $j=1,2$, and for some $\gamma>0$ $$ \operatorname{Var}\omega_2(x)|_y^\infty=O(e^{-y^{1+\gamma}}),\quad y\to\infty; $$ 3) $\int_{-\infty}^\infty e^{yx}dV(x)\ne0,\quad-\infty$. In the paper the following result is obtained Theorem. If $V_1(x)$ and $V_2(x)\in B$ and $V_1*V_2=\Phi$, where $\Phi$ is a normal distribution function, then $V_1$ and $V_2$ are normal (may be degenerate).