Might there be a probability distribution $\mathcal{D}$ such that if we sample $a_i \sim \mathcal{D}([-N,N])$ where $[-N,N] \subset \mathbb{Z}$ then if we define the asymptotic estimate $f$:

\begin{equation} P(\sum_{i=1}^{2n} a_i =0 ) \sim f(n) \end{equation}

$f$ is an oscillating function? My intuition suggests that only contrived examples might exist but my intuition has been wrong in the past.

I'm particularly interested in examples that occur in applied mathematics.