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Teoriya Veroyatnostei i ee Primeneniya, 1981, Volume 26, Issue 4, Pages 847–857
(Mi tvp3517)
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This article is cited in 8 scientific papers (total in 8 papers)
Short Communications
On limit theorems on large deviations in narrow zones
L. V. Rozovskiĭ Leningrad
Abstract:
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
$$
\Lambda(z)/z\uparrow\infty,\quad\Lambda(z)/z^{1+\varepsilon}\downarrow 0\quad(0<\varepsilon<1,\ z>z_0),
$$
the sequence $B_n\to\infty$ ($n\to\infty$) and
$$
\sup_{x\ge 0}|\mathbf P\{S_n<xB_n\}-\Phi(x)|=o(1),\qquad n\to\infty.
$$
Received: 03.01.1979
Citation:
L. V. Rozovskiǐ, “On limit theorems on large deviations in narrow zones”, Teor. Veroyatnost. i Primenen., 26:4 (1981), 847–857; Theory Probab. Appl., 26:4 (1982), 834–845
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https://www.mathnet.ru/eng/tvp3517 https://www.mathnet.ru/eng/tvp/v26/i4/p847
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