The purpose of this introductory presentation is to derive limit theorems for a class of random walks on random point processes, which mathematically describe the motion of a particle on a random environment. 

In the physical literature, the continuous-time version of this process is called Lévy-Lorentz gas. 

This model embodies the features of many different natural systems that show the so-called superdiffusive behavior, from particles moving in a turbulent fluid to the motion of crawling amoeba and the foraging strategies of several animals. 

We will start by recalling some classic results and definitions regarding the distributional convergence of stochastic processes, then we will move to the presentation of our model, starting from the definition of a one-dimensional random environment. 

We will finally present the convergence results for the process Y which takes place on such environment and that under suitable hypotheses displays a super-diffusive behavior. In particular, we will determine the scale whereby Y converges to a non-null limit and the limit process to which it converges.