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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.

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Can you be a bit more specific about what you want to hear about this distribution? I mean, we can certainly say something, but if you could indicate the direction in which you are looking, we could try to look in the same direction when answering :). –  fedja Aug 30 '12 at 3:09
    
can you use reflection principle? I think you have to define $X_i = |X_{i-1}+A_i|$ in order to say 0 is reflecting. –  john mangual Aug 30 '12 at 6:55
    
Poisson random variables $A_i$ don't take negative values, so in that case $X_i=X_{i-1}+A_i$ and so $X_i=A_1+\cdots +A_i$ itself has a Poisson distribution. –  Byron Schmuland Sep 1 '12 at 14:17

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