  We consider the convergence of a continuous-time Markov chain approximation
X^h, h>0, to an R^d-valued Levy process X. The state space of X^h is an
equidistant lattice and its Q-matrix is chosen to approximate the generator of
X. In dimension one (d=1), and then under a general sufficient condition for
the existence of transition densities of X, we establish sharp convergence
rates of the normalised probability mass function of X^h to the probability
density function of X. In higher dimensions (d>1), rates of convergence are
obtained under a technical condition, which is satisfied when the diffusion
matrix is non-degenerate.
