Geodesic
On determining an infinitely divisible distribution function by its values on a half-line
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 393-399 
I. A. Ibragimov. On determining an infinitely divisible distribution function by its values on a half-line. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 393-399. http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a15/
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     author = {I. A. Ibragimov},
     title = {On determining an infinitely divisible distribution function by its values on a~half-line},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {393--399},
     year = {1977},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a15/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Theorem. {\it Let $F(x)$ be an infinitely divisible distribution function with characteristic function $f(t)$. Suppose $f$ is holomorphic in $\{\operatorname{Im} z>0\}$ ($\{\operatorname{Im} z<0\}$). If an infinitely divisible distribution function $G$ coincides with $F$ on a half-line $(-\infty,a)$ (on a half-line $(a,\infty)$) then either $F(x)$ equals zero (equals one) on the half-line or $F(x)=G(x)$ for all $x$.} The theorem generalizes a result of H. Rossberg [1]. Examples are given which show that the analiticity condition is essential.