Densities with spherical level sets in the Gauss - exponential domain

For a sequence of independent observations Zn = (Xn, Yn) from the bivariate standard normal density there are at least three asymptotic descriptions of the sample clouds: • Let rn = √ 2 log n. The sample clouds {Z1/rn, . . . ,Zn/rn} converge onto the closed unit disk E. The probability of a sample point outside a disk of radius r > 1 tends to zero, as does the probability for any m ≥ 1 of less than m sample points in a disk centered in a point of E. • The two components of Z1 are independent, and hence asymptotically independent. The margins lie in the domain of the Gumbel distribution. Hence the sample clouds converge in distribution to a Poisson point process on the space X = [−∞,∞] \ {(−∞,−∞)}. The mean measure lives on the two boundary lines in −∞. • Let Hn be halfplanes such that P{Z ∈ Hn} = 1/n. The high risk scenario Zn describes the vector Z conditioned to lie in the halfplane Hn. There exist affine transformations αn mapping the upper halfplane {v ≥ 0} ontoHn such that α−1 n (Zn)⇒W = (U, V ). The components U and V are independent variables, Gaussian and exponential. Under the same normalization the sample clouds converge in distribution to a Poisson point process N on the plane with Gauss-exponential intensity. This paper looks at sample clouds from light-tailed unimodal densities with spherical level sets. What conditions will give the asymptotic behaviour above? We do not assume that the level sets are concentric.