Limit theorems for polylinear forms and quasi-polynomial functions

V. I. Rotar'

Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 3, pp. 527-545

Résumé

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 $n\to\infty$, $$ \mathbf P_F(\zeta_n)-\mathbf P_G(\zeta_n)\to0 $$ for almost all $x$ with respect to the Lebesgue measure on $R^1$.