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

Abstract

We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: 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 $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.