Geodesic
On the Chebyshev–Cramér asymptotic expansions
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 506-519 
I. A. Ibragimov. On the Chebyshev–Cramér asymptotic expansions. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 506-519. http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a8/
@article{TVP_1967_12_3_a8,
     author = {I. A. Ibragimov},
     title = {On the {Chebyshev{\textendash}Cram\'er} asymptotic expansions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {506--519},
     year = {1967},
     volume = {12},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_3_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\xi_1,\dots,\xi_n,\dots$ be a sequence of independent identically distributed random variables with a distribution function (d.f.) $F(x)$ and let $\mathbf E\xi_i=0$, $\mathbf D\xi_i=1$. Denote $\mathbf P\Bigl\{\frac1{\sqrt n}\sum_1^n\xi_i. Let $\beta_1,\beta_2,\dots,\beta_n,\dots$ be a numerical sequence such that $\beta_1=\mathbf E\xi_1=0$, $\beta_2=\mathbf E\xi_1^2=1$ and the other $\beta_s$ are arbitrary. Let us connect with the $\beta$-sequence the sequence $\{Q_n(x)\}$ of the Chebyshev–Cramér polynomials constructed in such a way as if $\{\beta_n\}$ were the sequence of moments of some distribution. We investigate the rate of convergence of the difference $$ \sup\limits_n\biggl|F_n(x)-\biggl[\Phi(x)+\frac1{\sqrt{2\pi}}e^{-x^2/2}\sum_{s=1}^k\frac{Q_s(x)}{n^{s/2}}\biggr]\biggr| $$ to zero (here $\Phi(x)$ is the normal d.f.).