A Superdiffusive Local Limit Theorem for the Elephant Random Walk and the Breakdown of Log-Concavity

Source abstract

Let $(S_n)_{n\geq 1}$ be the one-dimensional Elephant Random Walk (ERW) in the superdiffusive regime, with memory parameter $p\in(3/4,1)$ and first-step bias $\mathbb{P}(S_1=1)=q$. Set $a=2p-1\in(1/2,1)$. If $f_{q,a}$ denotes the density of the superdiffusive limit $L_{q,a}$, we prove the uniform local limit theorem $\lim_{n\to\infty}\sup_{j\in\mathbb{Z},\,j\equiv n\;(\mathrm{mod}\;2)}\left|\frac{n^a}{2}\mathbb{P}(S_n=j)-f_{q,a}(j/n^a)\right|=0$. The proof uses the exact recurrence for the probability mass function of $S_n$, yielding uniform $O(n^{-a})$ and $O(n^{-2a})$ bounds for the p.m.f. and its first discrete differences, respectively. We then disprove a conjecture of Guérin, Laulin, Raschel and Simon concerning eventual log-concavity. Writing $L_a=L_{1,a}$ and $f_a=f_{1,a}$, let $a_{\rm ev}$ be the eventual-log-concavity threshold and let $a_\star$ be the upper failure threshold for log-concavity of $f_a$. Their results imply $a_{\rm ev}\geq(\sqrt{5}-1)/2$, and we prove $(\sqrt{5}-1)/2\leq a_{\rm ev}\leq\min\{0.80399,a_\star\}\leq a_\star<0.918$. The bound $a_\star<0.918$ follows from the first three exact moments of $L_a$ and the sharp skewness inequality for centered log-concave distributions, while $a_{\rm ev}\leq0.80399$ follows from a certified finite-order analysis of the exact p.m.f. recurrence. Direct numerical iterations provide evidence that $a_{\rm ev}\approx0.80399$, equivalently $p_{\rm ev}\approx0.902$.