$L^p$-Boundedness of the Overshoot in Multidimensional Renewal Theory

Abstract

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random variables leaves a ball of radius $r$ in some given norm on $\mathbb{R}^d$. In the case of the Euclidean norm we completely characterize $L^p$-boundedness of the overshoot $\|S_{T_r}\| - r$ in terms of the underlying distribution. For more general norms we provide a similar characterization 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 characterization of the possible limit laws of $S_{T_r}/\|S_{T_r}\|$ under the weaker condition $\|S_{T_r}\|/r \rightarrow_p 1$.