Limit Theorems for Random Multinomial Forms

Alfredas Basalykas
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Alfredas Basalykas: Inst. of Mathem. and Inform.

A chapter in Approximation, Probability, and Related Fields, 1994, pp 101-107 from Springer

Abstract: 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 n (k) 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 n (k) reduces to the multilinear form η n (k) of order k $$ \eta _n^{\left( k \right)} = \eta _n^{\left( k \right)} \left( {X_1 , \ldots ,X_n } \right) = n^{ - k/2} {\text{ }}\sum\limits_{1 \leqslant i_1 \ne \ldots \ne i_k n} {\text{ }} a_{i_1 , \ldots ,i_k }^{\left( n \right)} X_{i_1 } \ldots X_{i_k } . $$

Date: 1994
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DOI: 10.1007/978-1-4615-2494-6_6

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