Generalized domains of semistable attraction of nonnormal laws

M.M. Meerschaert and H.-P. Scheffler

Abstract: Operator semistable laws are the natural multivariable analogue of semistable laws in one variable. Operator semistable laws occur as the limit of normalized and centered sums of i.i.d. random vectors when we consider only the sums which terminate at some $k_n$, with the ratio of successive $k_n$ tending to some constant $c\geq 1$. The generalized domain of semistable attraction of an operator semistable law consists of all such underlying distributions, when we allow normalizing by linear operators. In this paper we give concise necessary and sufficient conditions for a probability distribution to belong to the generalized domain of semistable attraction of any operator semistable law having no normal component. These results, together with the case of a normal limit, constitute a more general framework in which an i.i.d. sum of random vectors can be usefully approximated by a limit distribution. We anticipate a number of applications to multivariate analysis for random vectors whose covariance matrix is undefined because of heavy tails.

Keywords: Operator semistable laws,generalized domains of attraction, regular variation

Classification (MSC2000): 60F05; 26B35

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