Geodesic
Unimprovability of the result due to N.\,A.~Sapogov in the stability problem of Cram\'er's theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 522-525

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We consider the sequence (1) of compositions of distribution functions satisfying the condition (2). Let truncated variances of components be bounded from below by a positive constant. It is proved that the well-known estimate $$ \max_{i=1,2}\inf_{G\in N}\sup_x|F_n^{(i)}(x)-G(x)|=O\biggl(\frac1{\sqrt{-\ln\varepsilon_n}}\biggr) $$ (where $N$ is the set of all normal distribution functions) is unimprovable. 
@article{TVP_1968_13_3_a14,
     author = {S. G. Maloshevskii},
     title = {Unimprovability of the result due to {N.\,A.~Sapogov} in the stability problem of {Cram\'er's} theorem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {522--525},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a14/}
}
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S. G. Maloshevskii. Unimprovability of the result due to N.\,A.~Sapogov in the stability problem of Cram\'er's theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 522-525. http://geodesic.mathdoc.fr/item/TVP_1968_13_3_a14/