Symmetric Markov chains on $\mathbb{Z}^d$ with unbounded range

We consider symmetric Markov chains on $\mathbb{Z}^d$ where we do not assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on $\mathbb{R}^d$ corresponding to an elliptic operator in divergence form.