On the regular variation of elliptical random vectors

Let S = (S-1,...,S-d)(T), d >= 2 be a spherical random vector in R-d and let X = A(inverted perpendicular)S be an elliptical random vector with A is an element of R-dxd a non-singular matrix. Berman (1992. Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/ Cole) proved that if the random radius R-d = (Sigma(d)(i=1) S-i(2))(1/2) regularly varying with index alpha > 0 then S and S-i, 1 <= i <= d are regularly varying with index alpha. In this paper we derive several new equivalent conditions for the regular variation of X. As a by-product we obtain two asymptotic results concerning the sojourn and supremum of Berman processes. (C) 2006 Elsevier B.V. All rights reserved.