Let $P_i$, $N_i$, and $Z_i$, $i\in\mathbb{N}$ be r.v.'s with the $P_i$, $N_i$, and $Z_i$ being identically distributed with known pdf's $f_P$, $f_N$, and $f_Z$, respectively; and with no dependence between any $P_i$, $N_i$, or $Z_i$. Define the following recurrent process:

$$T_0:=0$$

$$T_i:=\left\{ \begin{array}{ll} Z_i & \quad T_{i-1} = 0 \\ min(T_{i-1}+N_i, 0) & \quad T_{i-1} < 0 \\ max(T_{i-1}+P_i, 0) & \quad T_{i-1} > 0 \end{array} \right.$$

In English: a particle walks randomly on the real number line (starting from the origin) in discrete time steps, where its position cannot change sign from positive to negative or vice-versa in a single step - were it to do so, it is set back at the origin before the next step - and where its evolution depends on the sign of its current position.

What can be said about the pdf's of the $T_i$ in relation to $f_P$, $f_N$, and $f_Z$?