Continuous-time random walk between L\'evy-spaced targets in the real line

We consider a continuous-time random walk which is defined as an interpolation of a random walk on a point process on the real line. The distances between neighboring points of the point process are i.i.d. random variables in the normal domain of attraction of an $\alpha$-stable distribution with $0 < \alpha < 1$. This is therefore an example of a random walk in a L\'evy random medium. Specifically, it is a generalization of a process known in the physical literature as L\'evy-Lorentz gas. We prove that the annealed version of the process is superdiffusive with scaling exponent $1/(\alpha + 1)$ and identify the limiting process, which is not c\`adl\`ag. The proofs are based on the technique of Kesten and Spitzer for random walks in random scenery.