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We study a random walk on a point process given by an ordered array of points$(\omega_k, \, k \in \mathbb{Z})$ on the real line. The distances $\omega_{k+1}- \omega_k$ are i.i.d. random variables in the domain of attraction of a$\beta$-stable law, with $\beta \in (0,1) \cup (1,2)$. The random walk hasi.i.d. jumps such that the transition probabilities between $\omega_k$ and$\omega_\ell$ depend on $\ell-k$ and are given by the distribution of a$\mathbb{Z}$-valued random variable in the domain of attraction of an$\alpha$-stable law, with $\alpha \in (0,1) \cup (1,2)$. Since the definingvariables, for both the random walk and the point process, are heavy-tailed, wespeak of a L\'evy flight on a L\'evy random medium. For all combinations of theparameters $\alpha$ and $\beta$, we prove the annealed functional limit theoremfor the suitably rescaled process, relative to the optimal Skorokhod topologyin each case. When the limit process is not c\`adl\`ag, we prove convergence ofthe finite-dimensional distributions. When the limit process is deterministic,we also prove a limit theorem for the fluctuations, again relative to theoptimal Skorokhod topology.

Limit theorems for L\'evy flights on a 1D L\'evy random medium