Limit Theorems for Random Multinomial Forms

Abstract

Let X ₁,X ₂,... be the sequence of i.i.d. (0,1) random variables with finite 2k moments for some integer k≥ 2. By Q ^(k)_n we denote the multinomial form of order k

$$ Q_n^{(k)} = Q_n^{(k)} \left( {X_1 ,...X_n } \right) = n^{ - k/2} \sum\limits_{1 \leqslant i_1 ,...,i_k \leqslant n} {a_{i_1 ...i_k }^{(n)} X_{i_1 } ...X_{i_k } ,a_{i_1 ...i_k }^{(n)} \in R.} $$

When $\alpha _{{i_1} \ldots ,{i_k}}^{\left( n \right)} = 0$ if two or more indices coincide, then Q ^(k)_n reduces to the multilinear form η ^(k)_n of order k

$$ \eta _n^{\left( k \right)} = \eta _n^{\left( k \right)} \left( {X_1 , \ldots ,X_n } \right) = n^{ - k/2} {\text{ }}\sum\limits_{1 \leqslant i_1 \ne \ldots \ne i_k n} {\text{ }} a_{i_1 , \ldots ,i_k }^{\left( n \right)} X_{i_1 } \ldots X_{i_k } . $$