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Teor. Veroyatnost. i Primenen., 1973, Volume 18, Issue 1, Pages 109–121 (Mi tvp2684)  

This article is cited in 3 scientific papers (total in 3 papers)

On strengthening of Lyapunov type estimates (the case when summands distributions are close to the normal one)

S. V. Nagaeva, V. I. Rotar'b

a Novosibirsk
b Moscow

Abstract: Let $\{X_j\}_{j=1}^n$ be a sequence of independent random variables. Put
\begin{gather*} \mathbf MX_j=0,\quad\mathbf MX_j^2=\sigma_j^2,\quad B^2=\sum_{j=1}^n\sigma_j^2,\quad C=\sum_{j=1}^n\sigma_j^3;
\nu_j=3\int_{-\infty}^\infty x^2|F_j(x)-\Phi(x/\sigma_j)| dx \end{gather*}
where $F_j(x)=\mathbf P\{X_j<x\}$, $\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^xe^{u^2/2} du$. Let
$$ \Lambda=\sum_{j=1}^n\nu_j,\quad\delta=\sup_x|\mathbf P\{\sum_{j=1}^nX_j<Bx\}-\Phi(x)|. $$
In the paper, some estimates of $\delta$ are obtained. The simpliest consequence from these estimates is the following:
$$ \delta\le L\max\{\frac\Lambda{B^3};(\frac{\Lambda}{B^3})^{1/4}(\frac C{B^3})^{3/4}\} $$
where $L$ is an absolute constant.

Full text: PDF file (620 kB)

English version:
Theory of Probability and its Applications, 1973, 18:1, 107–119

Bibliographic databases:

Received: 04.03.1971

Citation: S. V. Nagaev, V. I. Rotar', “On strengthening of Lyapunov type estimates (the case when summands distributions are close to the normal one)”, Teor. Veroyatnost. i Primenen., 18:1 (1973), 109–121; Theory Probab. Appl., 18:1 (1973), 107–119

Citation in format AMSBIB
\Bibitem{NagRot73}
\by S.~V.~Nagaev, V.~I.~Rotar'
\paper On strengthening of Lyapunov type estimates (the case when summands distributions are close to the normal one)
\jour Teor. Veroyatnost. i Primenen.
\yr 1973
\vol 18
\issue 1
\pages 109--121
\mathnet{http://mi.mathnet.ru/tvp2684}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=324757}
\zmath{https://zbmath.org/?q=an:0284.60016}
\transl
\jour Theory Probab. Appl.
\yr 1973
\vol 18
\issue 1
\pages 107--119
\crossref{https://doi.org/10.1137/1118008}


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    Remarks

    This publication is cited in the following articles:
    1. V. I. Rotar', “On summation of independent variables in a non-classical situation”, Russian Math. Surveys, 37:6 (1982), 151–175  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. Theory Probab. Appl., 52:2 (2008), 361–370  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Paulauskas V., “On the rate of convergence to bivariate stable laws”, Lithuanian Mathematical Journal, 49:4 (2009), 426–445  crossref  mathscinet  zmath  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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