  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.
