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. 



Probability distributions with irregular behaviour
According to the local central limit theorem (see e.g. [Esseen, Theorem 5, page 63](https://projecteuclid.org/euclid.acta/1485888404)), the probability in question is 
$$\sim\frac d{2s\sqrt{\pi n}}
$$
as $n\to\infty$, where $s$ is the standard deviation of the distribution $\mathcal D$, provided the following conditions: (i) the mean of that distribution is $0$; (ii) $d$ is the GCD of $S-S$, where $S$ is the support set of $\mathcal D$; and (iii) $0$ is in the support set of the distribution of $\sum\_1^{2m}a\_i$ for some natural $m$. 


(The definition of the condition $(L\_d)$ used in the mentioned theorem by Esseen is given at the bottom of page 54 of Esseen's paper.)


