Geodesic
On the stability of decompositions of the unit distribution function
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 715-718

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Let $E$ be the unit distribution function $$ E(x)= \begin{cases} 0,\le0 \\ 1,>0 \end{cases} $$ and $F=F_1*F_2$ be a distribution function such that in the uniform metric $$ \rho(F,E)\le\varepsilon\le1/4. $$ Let $F_1$ have median 0. We show that $$ \rho(F_1,E)\le\frac{1-\sqrt{1-4\varepsilon}}2. $$ and this estimate can not be improved. 
@article{TVP_1969_14_4_a9,
     author = {J. Macys},
     title = {On the stability of decompositions of the unit distribution function},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {715--718},
     publisher = {mathdoc},
     volume = {14},
     number = {4},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_4_a9/}
}
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J. Macys. On the stability of decompositions of the unit distribution function. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 715-718. http://geodesic.mathdoc.fr/item/TVP_1969_14_4_a9/