Geodesic
Estimations in the theorem of the stability of Poisson distribution decompositions
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 218-228

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Let $\Pi_\lambda$ be a Poisson distribution function and $F=F_1*F_2$ a distribution function such that either in the Lévy metric or in the uniform metric $\rho(F,\Pi_\lambda)\le\varepsilon$. We show that, there exists a Poisson distribution function $\Pi_{\lambda_1}$ such that $$ \rho(F_1,\Pi_{\lambda_1})(\lambda)\sqrt{\frac{\ln(-\ln\varepsilon)}{(-\ln\varepsilon)}}. $$ 
@article{TVP_1971_16_2_a1,
     author = {J. J. Ma\v{c}ys},
     title = {Estimations in the theorem of the stability of {Poisson} distribution decompositions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {218--228},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a1/}
}
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J. J. Mačys. Estimations in the theorem of the stability of Poisson distribution decompositions. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 2, pp. 218-228. http://geodesic.mathdoc.fr/item/TVP_1971_16_2_a1/