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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,\dots,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}\dots X_{j_k},
\end{gather*}
where
$\bar j=\{j_1,\dots,j_k\}$ be a sample from
$(1,\dots,n)$,
\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,\dots\},
\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
$R^1$.
Received: 26.09.1974
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