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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1973, Volume 18, Issue 3, Pages 527–534 (Mi tvp2725)

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)}|+[\sum_{i,j,i\ne j}(a_{ij}^{(n)})^2]^{1/2},
e_{jn}^2=|a_{jj}^{(n)}|+[\sum_k ^{(j)}(a_{kj}^{(n)})^2]/b_n^2(\sum_i ^{(j)}a_i=\sum_ia_i-a_j). \end{gather*}

Here is a typical result of the paper. Theorem {\em 1. Let $X^{(n)}=(X_1,…,X_n)$, $\zeta_n=(A_nX^{(n)},X^{(n)})b_n^2$. 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}.

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English version:
Theory of Probability and its Applications, 1974, 18:3, 499–507

Bibliographic databases:

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://www.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} 

• http://mi.mathnet.ru/eng/tvp2725
• http://mi.mathnet.ru/eng/tvp/v18/i3/p527

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Gotze F., Tikhomirov A.N., “Asymptotic distribution of quadratic forms”, Annals of Probability, 27:2 (1999), 1072–1098
2. Roellin A., “Stein's Method in High Dimensions with Applications”, Ann. Inst. Henri Poincare-Probab. Stat., 49:2 (2013), 529–549