  Continuous time random walks (CTRWs) are versatile models for anomalous
diffusion processes that have found widespread application in the quantitative
sciences. Their scaling limits are typically non-Markovian, and the computation
of their finite-dimensional distributions is an important open problem. This
paper develops a general semi-Markov theory for CTRW limit processes in
$\mathbb{R}^d$ with infinitely many particle jumps (renewals) in finite time
intervals. The particle jumps and waiting times can be coupled and vary with
space and time. By augmenting the state space to include the scaling limits of
renewal times, a CTRW limit process can be embedded in a Markov process.
Explicit analytic expressions for the transition kernels of these Markov
processes are then derived, which allow the computation of all finite
dimensional distributions for CTRW limits. Two examples illustrate the proposed
method.
