Convergence of Scaled Random Samples in Rᵈ

Let \Xⱼ, 1 j n\ be a sequence of iid random vectors in Rᵈ and Sₙ = \Xⱼ/bₙ, 1 j n\. When do there exist scaling constants bₙ such that Sₙ converges to some compact set S in Rᵈ almost surely (in probability)? We show that a limit set S is star-shaped (i. e. , x S implies tx S, for 0 t 1) so that after a polar coordinate transformation the limit set is the hypograph of an upper semicontinuous function. We specify necessary and sufficient conditions for convergence to a particular limit set. Some examples are also given.