On the Chebyshev--Cram\'er asymptotic expansions

I. A. Ibragimov

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 506-519

Résumé

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.).