L_p-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

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By this author: Mynbaev K.T.

Author: Mynbaev K.T.

Source: Advances in Applied Mathematics, Volume 26, Number 4, May 2001 , pp. 302-329(28)

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Abstract:

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method. Copyright 2001 Academic Press.