I have a sum $T^i$ of zero/one Bern$(p)$ random variables $T_i$ and multiple disjoint absorbing regions, i.e. the absorbing region is a union of disjoint, closed sets:
$$T^i \in \bigcup_{u \in \mathbb{N}} [ud_i - t_i^*, ud_i+t_i^*]$$
where $d_i = i+1$ and $t_i^*$ is some complicated function that has the property that $t_i^* < 0$ for the first few $i$:s and then $t_i^*>0$ and grows with $i$. How would I go about calculating the expected first time of passage here? Due to starting out with $t_i < 0$, $T^i$ may end up between any of these "strips".
I was considering introducing a new variable representing the distance to the closest strips and exploiting the fact that all strips are simply shifted copies, so it doesn't really matter which strip you start in - all that matters is the distance to the closest boundaries, as long as $t_i^*>0$. Is this the only way?