The estimates of stability of the characterization of the normal distribution given by G.~Polya theorem

R. V. Yanushkevichius

Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 196-205

Résumé

Let $X_1$, $EX_1=0$ and $X_2$ be independent identically distributed random variables and let $$ \sup_x|P(X_1)-P(aX_1+bX_2)|\leq\varepsilon, $$ 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 $$ \sup_x|P(X_1)-\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2\sigma^2}\, dy|\leq c(\sigma^{-2p}+\sigma^{r-2p})\varepsilon^{1-2p/r}. $$