Snakes and perturbed random walks

We study some properties of random walks perturbed at extrema, which are generalizations of the walks considered, e.g., by Davis (1999) and Tóth (1996). This process can also be viewed as a version of an excited random walk, recently studied by many authors. We obtain several properties related to the range of the process with infinite memory and prove the strong law, the central limit theorem, and the criterion for the recurrence of the perturbed walk with finite memory. We also state some open problems. Our methods are predominantly combinatorial and do not involve complicated analytic techniques.