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Zapiski Nauchnykh Seminarov LOMI, 1980, Volume 97, Pages 6–14
(Mi znsl3259)
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This article is cited in 3 scientific papers (total in 3 papers)
On narrow domains of the integral normal convergence
N. N. Amosova
Abstract:
Let $X_{n1},\dots,X_{nk_n}$ be independently distributed random values with distribution functions
$F_{n1}(x),\dots,F_{nk_n}(x)$, $n=1,2,\dots$. Let $c>0$ and let
$$
\int_{-\infty}^\infty x\,dF_{ni}(x)=0,\quad
\int_{-\infty}^\infty x^2\,dF_{ni}(x)=\sigma^2_{ni}<\infty.
$$
Put
$$
B_n^2=\sum_{i=1}^n\sigma_{ni}^2,\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt,\quad\log_mz=\underbrace{\log\log\dots\log}_{m\text{ раз}}z,\quad m\ge1.
$$
Sufficient conditions are found for the relations to hold
\begin{gather}
P\biggl(\frac{X_{n1}+\dots+X_{nk_n}}{B_n}\ge x\biggr)=(1-\Phi(x))(1+0(1)),
\quad n\to\infty,\notag\\
P\biggl(\frac{X_{n1}+\dots+X_{nk_n}}{B_n}<-x\biggr)=\Phi(x)(1+0(1)),
\quad n\to\infty,
\notag
\end{gather}
univormly in $x\in[0,c\sqrt{\log_mB_n^2}]$, $m\ge1$.
Citation:
N. N. Amosova, “On narrow domains of the integral normal convergence”, Problems of the theory of probability distributions. Part VI, Zap. Nauchn. Sem. LOMI, 97, "Nauka", Leningrad. Otdel., Leningrad, 1980, 6–14; J. Soviet Math., 24:5 (1984), 483–489
Linking options:
https://www.mathnet.ru/eng/znsl3259 https://www.mathnet.ru/eng/znsl/v97/p6
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