  In this paper continuous time random walk models approximating fractional
space-time diffusion processes are studied. Stochastic processes associated
with the considered equations represent time-changed processes, where the
time-change process is a L\'evy's stable subordinator with the stability index
$\beta \in (0,1).$ In the parer the convergence of constructed CTRWs to
time-changed processes associated with the corresponding fractional diffusion
equations are proved using a new analytic method.
