On the regular variation of elliptical random vectors

Let S=(S1,...,Sd)[inverted perpendicular],d[greater-or-equal, slanted]2 be a spherical random vector in and let X=A[inverted perpendicular]S be an elliptical random vector with a non-singular matrix. Berman (1992. Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole) proved that if the random radius is regularly varying with index [alpha]>0 then S and Si,1[less-than-or-equals, slant]i[less-than-or-equals, slant]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.Regularly varying vectors Elliptical random vectors Berman process Sojourn limit theorem Asymptotics of supremum Coefficient of upper tail dependence