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 $$ \sup_x|P(X_1<x)-P(aX_1+bX_2<x)|\leq\varepsilon, $$ 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 $$ \sup_x|P(X_1<x)-\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2\sigma^2}\, dy|\leq c(\sigma^{-2p}+\sigma^{r-2p})\varepsilon^{1-2p/r}. $$