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.

  • $\begingroup$ 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 $n$. $\endgroup$
    – Matt F.
    Aug 28 '19 at 14:38
  • $\begingroup$ @MattF. Sorry, I forgot to add that this must be a sum of an even number of variables. $\endgroup$ Aug 28 '19 at 15:12
  • 2
    $\begingroup$ 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 $3 \mid n$. $\endgroup$
    – Matt F.
    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.)


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