Markov chain approximations for symmetric jump processes

Markov chain approximations of symmetric jump processes are investigated. Tightness results and a central limit theorem are established. Moreover, given the generator of a symmetric jump process with state space $\mathbbm{R}^d$ the approximating Markov chains are constructed explicitly. As a byproduct we obtain a definition of the Sobolev space $H^{\alpha/2}(\mathbbm{R}^d)$, $\alpha \in (0,2)$, that is equivalent to the standard one.