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Teoriya Veroyatnostei i ee Primeneniya, 1973, Volume 18, Issue 3, Pages 527–534 (Mi tvp2725)  

This article is cited in 23 scientific papers (total in 23 papers)

Some limit theorems for polynomials of second order

V. I. Rotar'

Moscow
Abstract: Let $X_i$ be independent identically distributed random variables, $P_F(E)$ the probability of a set $E$, when the distribuion function of $X_i$ is $F$, class $\mathscr F_{0,1}=\{F\colon x\,dF(x)=0;\ \int x^2dF(x)=1\}$. Let $A_n=\{a_{ij}^{n}\}$ be a $(n\times n)$ symmetric matrix,
\begin{gather*} b_n^2=\sum_i|a_{ii}^{(n)}|+\biggl[\sum_{i,j,i\ne j}(a_{ij}^{(n)})^2\biggr]^{1/2}, \\ e_{jn}^2=|a_{jj}^{(n)}|+\biggl[\sum_k{}^{(j)}(a_{kj}^{(n)})^2\biggr]\bigg/b_n^2\biggl(\sum_i{}^{(j)}a_i=\sum_ia_i-a_j\biggr). \end{gather*}

Here is a typical result of the paper. Theorem {\em 1. Let $X^{(n)}=(X_1,\dots,X_n)$, $\zeta_n=(A_nX^{(n)},X^{(n)})b_n^2$. Then, if $\max\limits_je_{jn}^2=o(b_n^2)$, for any $F,G\in\mathscr F_{0,1}$
$$ \mathbf P_F(\zeta_n<x)-\mathbf P_G(\zeta_n<x)\underset{n\to\infty}\longrightarrow0 $$
for any $x$, possibly excluding $x$ from a set of zero Lebesgue measure}.
Received: 25.11.1971
English version:
Theory of Probability and its Applications, 1974, Volume 18, Issue 3, Pages 499–507
DOI: https://doi.org/10.1137/1118064
Bibliographic databases:
Language: Russian
Citation: V. I. Rotar', “Some limit theorems for polynomials of second order”, Teor. Veroyatnost. i Primenen., 18:3 (1973), 527–534; Theory Probab. Appl., 18:3 (1974), 499–507
Citation in format AMSBIB
\Bibitem{Rot73}
\by V.~I.~Rotar'
\paper Some limit theorems for polynomials of second order
\jour Teor. Veroyatnost. i Primenen.
\yr 1973
\vol 18
\issue 3
\pages 527--534
\mathnet{http://mi.mathnet.ru/tvp2725}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=326803}
\zmath{https://zbmath.org/?q=an:0304.60037}
\transl
\jour Theory Probab. Appl.
\yr 1974
\vol 18
\issue 3
\pages 499--507
\crossref{https://doi.org/10.1137/1118064}
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  • https://www.mathnet.ru/eng/tvp/v18/i3/p527
  • This publication is cited in the following 23 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Теория вероятностей и ее применения Theory of Probability and its Applications
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