Harnack inequality for subordinate random walks

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice ${\mathbb{Z}}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.