I am not very familiar with the theory of martingales or random walks, perhaps someone could point me in the right direction or give me some help with the following problem.
Consider a random variable $X_t$, for $t = 0, 1, \dots$. We start the walk at $X_0 = 0$, and each step has $X_i = \max(0, X_{i-1} + A_i)$, where $A_i$ are independent, mean-zero, bounded random variables. This is basically a martingale, except there is a reflecting barrier at zero.
What can one say about the distribution of $X_t$?
In one case I am looking at, say $A_i$ are iid distributed as $Poisson(\lambda) - \lambda$. Does this make the analysis more tractable?
Thanks for any help.