Some limit theorems for polynomials of second order

V. I. Rotar'

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 3, pp. 527-534

Résumé

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)}|+\biggl[\sum_{i,j,i\ne j}(a_{ij}^{(n)})^2\biggr]^{1/2}, \\ e_{jn}^2=|a_{jj}^{(n)}|+\biggl[\sum_k{}^{(j)}(a_{kj}^{(n)})^2\biggr]\bigg/b_n^2\biggl(\sum_i{}^{(j)}a_i=\sum_ia_i-a_j\biggr). \end{gather*} Here is a typical result of the paper. Theorem {\em 1. Let $X^{(n)}=(X_1,\dots,X_n)$, $\zeta_n=(A_nX^{(n)},X^{(n)})b_n^2$. 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}.