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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
\begin{gather*} X_j\in R^1,\quad j=1,…,n,\quad\mathbf EX_j=0,\quad\mathbf EX_j^2=1
\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);
F_j(A)=\mathbf P(X_j\in A),\quad F=\{F_1,F_2,…\}, \end{gather*}
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:

Received: 26.09.1974

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}


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