On the position of a random walk at the time of first exit from a sphere (Q1196942)

Let \(X_ 1,X_ 2,\dots\) be independent and identically distributed \(d\)- dimensional random vectors with partial sums \(S_ n=\sum_{i=1}^ n X_ i\). Let \(T_ r\) be the first exit time of the random walk \((S_ n)\) from the \(\|\cdot\|\)-sphere with radius \(r\); here \(\|\cdot\|\) denotes an arbitrary norm on \(\mathbb{R}^ d\). The authors investigate the behaviour of the `radial overshoot' \(\| S_{T_ r}\|-r\) as \(r\to\infty\). Various criteria, essentially in terms of the distribution of \(X_ 1\), are obtained for growth rates and boundedness of moments of this quantity.