Markov Chain Approximation of Pure Jump Processes
  In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains. 
Ante Mimica, Nikola Sandri\'c, and Ren\'e L. Schilling
math.PR