Geodesic
On the problem of the stability of the decomposition of the normal law into components
Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 738-742

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The following theorem is proved: Let $\Phi$ be the distribution function of $N(0,1)$, $L$ the Levy metric and $F=F_1*F_2$ a distribution function such that $$ L(F,\Phi)\le\varepsilon1. $$ Then, there can be found a normal distribution $\Phi_1$ such that $$ C_1\biggl(\log\frac1\varepsilon\biggr)^{-1/2}(F_1,\Phi_1)\biggl(\log\frac1\varepsilon\biggr)^{-1/11}, $$ where $C_1$ and $C_2$ are positive constants. 
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     author = {V. M. Zolotarev},
     title = {On the problem of the stability of the decomposition of the normal law into components},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {738--742},
     publisher = {mathdoc},
     volume = {13},
     number = {4},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1968_13_4_a14/}
}
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V. M. Zolotarev. On the problem of the stability of the decomposition of the normal law into components. Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 4, pp. 738-742. http://geodesic.mathdoc.fr/item/TVP_1968_13_4_a14/