This research will seek results and their applications of multivariate regular variation. The applications will be to least square estimators in multiple regression with random predictors. The notable part will be that the assumptions on the underlying distributions will be minimal, allowing for infinite moments and disparity among distributional types. Basic among problems of study is the regular variation of probability tails for product of a random vector with an independent scaler and for the matrix of cross products formed from a random vector. A related problem to be studied will involve the exponential tail behavior of convolutions. Extensions of the concepts of subexponentiality and convolution equivalency will be sought. The research will have applications for multivariate infinitely divisible laws for compound multivariate distributions.
