Record statistics and persistence for a random walk with a drift

Satya N. Majumdar, Gregory Schehr and Gregor Wergen

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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 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

Date: 2012-06, Revised 2012-08
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Published in J. Phys. A: Math. Theor. 45 (2012) 355002

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