Let $X_1, X_2, ...$ be i.i.d. random variables, $\mathbb{E} X_1 > 0$, and let $S_n = \sum\limits _{i = 1} ^n X_i$. Define $\tau = \max \{n \in \mathbb{N}: S_n \leq 0 \}$ with the convention $\tau = 0$ if $S_n > 0$ for all $n \in \mathbb{N}$.

Apart from the asymptotic bounds, is there something known about distribution of $\tau$, at least in some specific cases? I am interested in particular in the case $X_i = \xi _i - a$, where $\xi _1, \xi _2, ...$ are i.i.d. unit exponentials and $a \in (0,1)$. For the Brownian motion with drift the distribution of the last zero crossing is given in Theorem 2.1 in this paper. The settings in this question are a little bit similar to those here and here.