# Probability distributions with irregular behaviour

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.

• If you flip a $\pm1$ coin, and $f(n)$ is the probability of a zero sum after $n$ flips, then $f(n)$ oscillates between 0 for odd $n$ and positive values for even $n$. Aug 28 '19 at 14:38
• @MattF. Sorry, I forgot to add that this must be a sum of an even number of variables. Aug 28 '19 at 15:12
• If you flip a coin that can be $+2$ or $-1$, and $f(n)$ is the probability of a zero sum after $n$ flips, then evenness of $n$ does not matter, but $f(n)$ oscillates between 0 if $3 \nmid n$ and positive values if $3 \mid n$. Aug 28 '19 at 15:15

According to the local central limit theorem (see e.g. Esseen, Theorem 5, page 63), 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.)