On limit theorems on large deviations in narrow zones
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 847-857
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Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables, $S_n=X_1+\dots+X_n$, $\Phi(x)$ be the standard normal distribution function. We investigate the asymptotics of
$$
\mathbf P\{S_n>x\}/(1-\Phi(x/B_n)),\qquad n\to\infty,
$$
for $0\le x\le \Lambda(B_n)$, where the function $\Lambda(z)$ is such that
$$
\Lambda(z)/z\uparrow\infty,\quad\Lambda(z)/z^{1+\varepsilon}\downarrow 0\quad(0\varepsilon1,\ z>z_0),
$$
the sequence $B_n\to\infty$ ($n\to\infty$) and
$$
\sup_{x\ge 0}|\mathbf P\{S_n\}-\Phi(x)|=o(1),\qquad n\to\infty.
$$
@article{TVP_1981_26_4_a18,
author = {L. V. Rozovskiǐ},
title = {On limit theorems on large deviations in narrow zones},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {847--857},
publisher = {mathdoc},
volume = {26},
number = {4},
year = {1981},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a18/}
}
L. V. Rozovskiǐ. On limit theorems on large deviations in narrow zones. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 847-857. http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a18/