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 η drawn from a continuous distribution f (η), which is symmetric around a constant drift c . We focus in particular on the case where f (η) is a symmetric stable law with a Lévy index 0 < μ ⩽ 2. The record statistics crucially depends on the persistence probability, which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit of a large number of steps n , the record statistics is sensitive to these parameters ( c and μ) of the jump distribution. We compute the asymptotic mean record number 〈 R n 〉 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 < μ ⩽ 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.