Markov chain approximations for symmetric jump processes

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