Deterministic random walks driven by random resets

J. Villarroel, M. Montero Torralbo

We consider a random walk driven by a random reset mechanism that at certain random times refreshes the system to the starting value. This mechanism defines a renewal process. Between reset times the random walk increases deterministically. Natural objects to consider are statistics related to recurrence and extreme properties like first-visit times to a given level and escape and recurrence times. The complexity of the resulting generating functions implies that the corresponding distributions can not be inverted in closed form, even for the simplest choices of the parent cdf F. Nevertheless we prove that all states are positive recurrent and determine the mean recurrence time. We also relate the large distance behaviour with the tail of the cdf F. The distribution of the random measure counting the number of visits to a level or the number of delta records before a given time are determined. Formulas for the first-passage time and high-water marks are also derived.

Palabras clave: Random walk, Running maxima, Records, Rnewals

L05.4 Procesos Estocásticos I
5 de septiembre de 2016  11:30
Aula 21.07

E. Hernández Balaguera, H. Vara Rivera, J. L. Polo Sanz

M. González Velasco, C. Minuesa Abril, I. M. del Puerto García