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JOURNALS // Teoriya Veroyatnostei i ee Primeneniya // Archive

Teor. Veroyatnost. i Primenen., 1973 Volume 18, Issue 4, Pages 753–766 (Mi tvp4364)

This article is cited in 39 papers

On the rate of approach of the distributions of sums of independent random variables to accompanying distributions

I. A. Ibragimov, E. L. Presman


Abstract: Let $\mathcal{F}$ be the set of all distribution functions on $R,\mathcal{F}^*$ the subset of $\mathcal{F}$ corresponding to symmetric random variables, $F^n$ $n$-times convolution of $F$ with itself, $E_a$ the distribution function corresponding to the unit mass at $a$, $|F-G|=\sup_x |F(x)-G(x)|$ for $F,G\in\mathcal{F}$.
It is proved that
$$ \frac{c_0}{n^{1/3}}<\sup_{F\in\mathcal{F}}\inf_a |(E_a F)^n-\exp\{n(E_a F-E_0)\}|\leq\frac{8}{n^{1/3}}, $$

$$ \frac{c_1}{\sqrt{n}}<\sup_{F\in\mathcal{F}^*}|F^n-\exp\{n(F-E_0)\}|<c_2\sqrt{\frac{\log n}{n}}. $$
Here the first right-hand inequality is Kolmogorov's uniform limit theorem in Le Cam's version.
We study also the closeness of distribution functions $\prod_i F_i E_{a_i}$ and $\exp\sum_i (F_iE_{a_i}-E_0)$ in the Kolmogorov–Smirnov and Lévy metrices.

Received: 05.10.1972


 English version:
Theory of Probability and its Applications, 1974, 18:4, 713–727

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