Let $x_1, x_2 \cdots x_M$ be a sequence of iid random variables taking values over the integers, with $E(x_i)=0$. In particular, I'm interested in a shifted Poisson: $X=P-1$, where $P$ is Poisson with $\lambda=1$.
Let $y_1, y_2 \cdots y_M$ be the partial sums, $y_i=\sum_{k=1}^i x_k$
Finally, let $z_M$ be their uniform $mixture$: the result of choosing randomly, with equal probability $1/M$, one of the $y_i$ values. So that
$p_{z_M}(t)=\frac{1}{M}\left[p_{y_1}(t)+p_{y_2}(t)+ \cdots +p_{y_M}(t)\right]$
and $E[z_M]=0$. I'm interested in the asympotics of $p_{z_M}$, $M\to \infty$; in particular, I'd wish a good (better than first order) approximation of $P(z_M = 0)$.
I arrived to this at attacking (without much success) this problem, and got me interested about the general behaviour of $z_M$. An Edgeworth-like expansion does not seem feasible here, $z_M$ does not tend to a gaussian - its kurtosis does not tend to zero. It's easy to see that the characteristic function of $z$ is given by
$Z(\omega) = \frac{X(\omega)}{M}\frac{1-{X(\omega)}^M}{1-X(\omega)}$
but that did not lead me very far. Another observation: if we regard $y_M$ as a random walk (with Poisson steps), $P(z_M = 0)$ would equal the probality of finding the trajectory at $y=0$ at some random time.