β Scribed by Kairat T. Mynbaev
No coin nor oath required. For personal study only.
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 squareintegrable 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 weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.
π SIMILAR VOLUMES
In this paper a series representation of the joint density and the joint distribution of a quadratic form and a linear form in normal variables is developed. The expansion makes use of Laguerre polynomials. As an example the calculation of the joint distribution of the mean and the sample variance i