Multivariable Domains of Attraction and Regular Variation.
Mark Marvin Meerschaert · Deep Blue (University of Michigan) · 1984
A sequence of the independent, identically distributed r and om vectors X(,1),X(,2),X(,3),..., is said to belong to the domain of attraction of a r and om vector Y if there exist linear operators A(,n) and constant, nonr and om vectors b(,n) such that the normalized sums A(,n) (X(,1) + ... + X(,n)) - b(,n) converge in distribution to Y. This dissertation addresses the problem of when there exist A(,n), b(,n), and Y such that the above holds for a given sequence of r and om vectors. The techniques used to solve this problem include a new multivariate theory of regular variation, the presentation of which constitutes a large part of the body of the dissertation. Along with the general domain of attraction problem two special cases are also considered in detail, in which the norming operators are replaced first by scalars, and then by vectors. Criteria in terms of regular variation for a given sequence of independent, identically distributed r and om vectors to belong to some domain of attraction are obtained in all three cases.