Ladder Costs for Random Walks in Lévy media

Abstract

We consider a random walk Y moving on a Lévy random medium, namely a one-dimensional renewal point process with inter-distances between points that are in the domain of attraction of a stable law. The focus is on the characterization of the law of the first-ladder height YT and length LT (Y ), where T is the first-passage time of Y in R + . The study relies on the construction of a broader class of processes, denoted Random Walks in Random Scenery on Bonds (RWRSB) that we briefly describe. The scenery is constructed by associating two random variables with each bond of Z, corresponding to the two possible crossing directions of that bond. A random walk S on Z with i.i.d increments collects the scenery values of the bond it traverses: we denote this composite process the RWRSB. Under suitable assumptions, we characterize the tail distribution of the sum of scenery values collected up to the first exit time T . This setting will be applied to obtain results for the laws of the first-ladder length and height of Y . The main tools of investigation are a generalized Spitzer-Baxter identity, that we derive along the proof, and a suitable representation of the RWRSB in terms of local times of the random walk S. All these results are easily generalized to the entire sequence of ladder variables.