Strong renewal theorems and local large deviations for multivariate random walks and renewals

We study a random walk Sn on Zd (d≥1), in the domain of attraction of an operator-stable distribution with index α=(α1,…,αd)∈(0,2]d: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function G(0,x) as ∥x∥→+∞, along the “favorite direction or scaling”: (i) if ∑di=1α−1i