Persistent Random Walks. II. Functional Scaling Limits

We describe the scaling limits of the persistent random walks (PRWs) for which the recurrence has been characterized in Cenac et al. (J. Theor. Probab. 31(1):232-243, 2018). We highlight a phase transition phenomenon with respect to the memory: depending on the tails of the persistent time distributions, the limiting process is either Markovian or non-Markovian. In the memoryless situation, the limits are classical strictly stable Levy processes of infinite variations, but the critical Cauchy case and the asymmetric situation we investigate fill some lacunae of the literature, in particular regarding directionally reinforced random walks (DRRWs). In the non-Markovian case, we extend the results of Magdziarz et al. (Stoch. Process. Appl. 125(11):4021-4038, 2015) on Levy walks (LWs) to a wider class of PRWs without renewal patterns. Finally, we clarify some misunderstanding regarding the marginal densities in the framework of DRRWs and LWs and compute them explicitly in connection with the occupation times of Lamperti's stochastic processes.