Record statistics and persistence for a random walk with a drift

TL;DR

This paper analyzes the record statistics and persistence of a biased one-dimensional random walk with symmetric stable jump distributions, revealing five distinct behavioral regions depending on the bias and jump distribution parameters.

Contribution

It provides exact asymptotic formulas for the mean record number, its distribution, and record age statistics for a biased random walk with stable Lévy jumps, identifying five behavioral regimes.

Findings

Record statistics depend on bias and Lévy index.

Five distinct behavioral regions identified.

Analytical results confirmed by simulations.

Abstract

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length \eta drawn from a continuous distribution f(\eta) which is symmetric around a constant drift c. We focus in particular on the case were f(\eta) is a symmetric stable law with a L\'evy index 0 < \mu \leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter \mu. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and \mu) 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 < \mu \leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.