L(P)-BOUNDEDNESS OF THE OVERSHOOT IN MULTIDIMENSIONAL, RENEWAL THEORY

Let T-r be the first time a sum S-n of nondegenerate i.i.d. random var iables leaves a ball of radius r in some given norm on R(d). In the ca se of the Euclidean norm we completely characterize LP-boundedness of the overshoot parallel to S(Tr)parallel to - r in terms of the underly ing distribution. For more general norms we provide a similar characte rization under a smoothness condition on the norm which is shown to be very nearly sharp. One of the key steps in doing this is a characteri zation of the possible limit laws of S-Tr/parallel to S(Tr)parallel to under the weaker condition parallel to S(Tr)parallel to/r-->(p) 1.