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On Asymptotically Best Constants in More Exact Forms of Mean Limit Theorems
V. M. Zolotarev Moscow
Abstract:
Let
$\mathfrak{B}$ be a class of distribution functions
$F$ with a finite third absolute moment
$\beta$ and first moment equal to zero. Also, let
$F_n (x)=F^{* n}(x\sigma\sqrt n)$, where
$\sigma^2 $ is the second moment
$F$,
$\Phi _{a,b}$ is the normal distribution with parameters
$(a,b)$ and
$\rho _3 $ is the mean metric, i.e., \[ \rho _3 (G,H)=\int|G - H|dx.\] C. Esseen [1] has calculated the value of
$$
A_3(F)=\lim_{n\to\infty}\sqrt n\rho _3(F_n,\Phi _{0,1}).
$$
In this paper we calculate the value of \[ \bar A_3 (F)=\mathop{\lim }\limits_{n \to\infty}\sqrt n\mathop{\inf }\limits_{a,b}\rho _3 (F_n,\Phi _{0,1}),\] the asymptotic of parameters
$a_n$,
$b_n$, which realize
$\inf.\rho _3$, and we prove that \[ B_3=\mathop{\sup }\limits_\mathfrak{B}\frac{{\sigma ^3 }}{\beta }A_3 (F)=\bar B_3=\mathop{\sup}\limits_\mathfrak{B} \frac{{\sigma ^3}}{\beta }\bar A_3 (F)=\frac{1}{2}.\]
Received: 16.12.1963