On the regular variation of elliptical random vectors

Let S=(S1,…,Sd)⊤,d⩾2 be a spherical random vector in Rd and let X=A⊤S be an elliptical random vector with A∈Rd×d a non-singular matrix. Berman (1992. Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole) proved that if the random radius Rd=(∑i=1dSi2)1/2 is regularly varying with index α>0 then S and Si,1⩽i⩽d are regularly varying with index α. 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.