Geodesic
The estimates of stability of the characterization of the normal distribution given by G. Polya theorem
Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 196-205 
R. V. Yanushkevichius. The estimates of stability of the characterization of the normal distribution given by G. Polya theorem. Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 196-205. http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a15/
@article{ZNSL_1979_87_a15,
     author = {R. V. Yanushkevichius},
     title = {The estimates of stability of the characterization of the normal distribution given by {G.~Polya} theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {196--205},
     year = {1979},
     volume = {87},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_87_a15/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $X_1$, $EX_1=0$ and $X_2$ be independent identically distributed random variables and let $$ \sup_x|P(X_1<x)-P(aX_1+bX_2<x)|\leq\varepsilon, $$ where $a>0$, $b>0$, $a^2+b^2=1$. Suppose that for some integer $r\geq3p$, $P=(1/2\ln(1/2))/\ln(\max(a,b))$, $$ \mathsf{E}|X_1|^r\leq M_r<\infty;\quad\mathsf{E}X_i^s=\mathsf{E}N_{0,\sigma}^s,\quad s=1,2,\dots,r-1, $$ where $N_{0,\sigma}$ – normal random variable with parameters $(0,\sigma)$. Then exist such constants $c=c(b,M_r)$ and $\varepsilon_0=\varepsilon_0(b,M_r,\sigma)$ that $$ \sup_x|P(X_1<x)-\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2\sigma^2}\, dy|\leq c(\sigma^{-2p}+\sigma^{r-2p})\varepsilon^{1-2p/r}. $$