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

 Teor. Veroyatnost. i Primenen., 1975, Volume 20, Issue 3, Pages 527–545 (Mi tvp3186)

Limit theorems for polylinear forms and quasi-polynomial functions

V. I. Rotar'

Moscow

Abstract: The paper deals with distributions of finite sets of polylinear forms and quasi-polynomial functions when the number of random arguments tends to infinity. As a particular case, arbitrary polynomials of random variables are considered.
The simplest corollary of our theorems is the following:
Let us consider random variables
\zeta_n=b_n^{-1}\sum_{\bar j}a(\bar j)X_{j_1}…X_{j_k}, \end{gather*}
where $\bar j=\{j_1,…,j_k\}$ be a sample from $(1,…,n)$,
\begin{gather*} b_n^2=\sum_{\bar j}a^2(\bar j);
let $\mathbf P_F(A)$ be the probability of $A$ for $F$, $\mathscr F$ be the class of $F$'s such that for any $F\in\mathscr F$ and $n\to\infty$
\begin{gather*} b_n^{-2}\sum_{j=1}^ns_j^2\int_{|x|>\varepsilon(b/s_j)^{1/k}}x^2F_j(dx)\to0,
s_j^2=\sum_{\bar j\ni j}a^2(\bar j). \end{gather*}
Then, for any $F$, $G\in\mathscr F$ and $n\to\infty$,
$$\mathbf P_F(\zeta_n<x)-\mathbf P_G(\zeta_n<x)\to0$$
for almost all $x$ with respect to the Lebesgue measure on $R^1$.

Full text: PDF file (1049 kB)

English version:
Theory of Probability and its Applications, 1976, 20:3, 512–532

Bibliographic databases:

Citation: V. I. Rotar', “Limit theorems for polylinear forms and quasi-polynomial functions”, Teor. Veroyatnost. i Primenen., 20:3 (1975), 527–545; Theory Probab. Appl., 20:3 (1976), 512–532

Citation in format AMSBIB
\Bibitem{Rot75} \by V.~I.~Rotar' \paper Limit theorems for polylinear forms and quasi-polynomial functions \jour Teor. Veroyatnost. i Primenen. \yr 1975 \vol 20 \issue 3 \pages 527--545 \mathnet{http://mi.mathnet.ru/tvp3186} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=385980} \zmath{https://zbmath.org/?q=an:0372.60030} \transl \jour Theory Probab. Appl. \yr 1976 \vol 20 \issue 3 \pages 512--532 \crossref{https://doi.org/10.1137/1120058}