Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in the presence of a constant bias c. At each time step, the walker makes a random jump of length. drawn from a continuous distribution f(eta), which is symmetric around a constant drift c. We focus in particular on the case where f (eta) is a symmetric stable law with a Levy index 0 after n steps as well as its full distribution P (R, n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < mu <= 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions