## The Annals of Probability

### $L^p$-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 $\|S_{T_r}\|/r \rightarrow_p 1$.

#### Article information

Source
Ann. Probab., Volume 23, Number 4 (1995), 2022-2056.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176987814

Digital Object Identifier
doi:10.1214/aop/1176987814

Mathematical Reviews number (MathSciNet)
MR1379179

Zentralblatt MATH identifier
0852.60084

JSTOR
Griffin, Philip S.; McConnell, Terry R. $L^p$-Boundedness of the Overshoot in Multidimensional Renewal Theory. Ann. Probab. 23 (1995), no. 4, 2022--2056. doi:10.1214/aop/1176987814. https://projecteuclid.org/euclid.aop/1176987814