  We consider a generalization of a one-dimensional stochastic process known in
the physical literature as L\'evy-Lorentz gas. The process describes the motion
of a particle on the real line in the presence of a random array of marked
points, whose nearest-neighbor distances are i.i.d. and long-tailed (with
finite mean but possibly infinite variance). The motion is a continuous-time,
constant-speed interpolation of a symmetric random walk on the marked points.
We first study the quenched random walk on the point process, proving the CLT
and the convergence of all the accordingly rescaled moments. Then we derive the
quenched and annealed CLTs for the continuous-time process.
