$L_p$-Approximable sequences of vectors and limit distribution of quadratic forms of random variables

The properties of $L_2$-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 $L_2$-approximable sequences since 1976 when Moussatat introduced a narrower notion of $L_2$-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 weights generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.