Consider a random walk $S_k = \sum_{i=1}^k X_i$ with i.i.d. increments from some continuous distribution $X$, with $E[X]=0$. Let $\tau_n$ be the first step $k$ for which $S_k > n$, for some threshold $n>0$. This occurs with probability 1. Let the "overshoot" $Z_n$ be defined as $Z_n=S_{\tau_n}-n$.
I am interested in the limiting overshoot distribution $Z=\lim_{n \to \infty} Z_n$.
What is known about $Z$ as a function of the increment distribution $X$? Are there special cases where $Z$ is known?
For instance, if $[X|X>0] \sim Exp(\mu)$, then $Z \sim Z_n \sim Exp(\mu), \forall n> 0$. Are there other results of this type?
If instead $X$ was a nonnegative distribution, then we could apply renewal theory to find $Z$. $Z$ would be the forward recurrence time (a.k.a. residual time, excess time), with density $f_Z(k) = \frac{1-F_X(k)}{E[X]}$. However, I don't know how to extend this analysis to allow $X$ to be negative.
Edit: In Asmussen, S., and V. Schmidt. "The ascending ladder height distribution for a certain class of dependent random walks." Statistica neerlandica 47.4 (1993): 269-277, the authors point out that if $X = A - B$, where $A$ and $B$ are independent random variables and $B$ is exponential, then $Z_0$ is equal to the residual time of $A$.
Moreover, by reading their references, it becomes clear that for any distribution $X$, $Z$ is simply the residual time of a renewal process with holding time $Z_0$. Thus in this specific case, $Z$ is simply the residual time of the residual time of $A$.