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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 (Greens function) up to order $mathcal{O} left( |x|^{frac{alpha-2}{3}+varepsilon} right)$ for any $varepsilon>0$ small enough, when $alpha in [1,2)$.
Given ${X_k}$ is a martingale difference sequence. And given another ${Y_k}$ which has dependency within the sequence. Assume ${X_k}$ is independent with ${Y_k}$, we study the properties of the sums of product of two sequences $sum_{k=1}^{n} X_k Y_k$
We consider a class of interacting particle systems with values in $[0,8)^{zd}$, of which the binary contact path process is an example. For $d ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the
We consider bootstrap percolation and diffusion in sparse random graphs with fixed degrees, constructed by configuration model. Every node has two states: it is either active or inactive. We assume that to each node is assigned a nonnegative (integer
We define a multi-group version of the mean-field spin model, also called Curie-Weiss model. It is known that, in the high temperature regime of this model, a central limit theorem holds for the vector of suitably scaled group magnetisations, that is