Local central limit theorem and potential kernel estimates for a class of symmetric heavy-tailted random variables
Year: 2021
Type: preprint
Abstract: In this article, we study a class of heavy-tailed random variables on $\mathbb{Z}$ in the domain of attraction of an $\alpha$-stable random variable of index $\alpha \in (0,2)$ satisfying a certain expansion of their characteristic function. Our results include sharp convergence rates for the local (stable) central limit theorem of order $n^{- (1+ \frac{1}{\alpha})}$, a detailed expansion of the characteristic function of a long-range random walk with transition probability proportional to $|x|^{-(1+\alpha)}$ and $\alpha \in (0,2)$ and furthermore detailed asymptotic estimates of the discrete potential kernel (Green's function) up to order $\mathcal{O} \left( |x|^{\frac{\alpha-2}{3}+\varepsilon} \right)$ for any $\varepsilon>0$ small enough, when $\alpha \in [1,2)$.
Source: arXiv (Cornell University)
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3118788232