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Record statistics and persistence for a random walk with a drift

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  • Satya N. Majumdar
  • Gregory Schehr
  • Gregor Wergen

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

Suggested Citation

  • Satya N. Majumdar & Gregory Schehr & Gregor Wergen, 2012. "Record statistics and persistence for a random walk with a drift," Papers 1206.6972, arXiv.org, revised Aug 2012.
    • Handle: RePEc:arx:papers:1206.6972
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    Cited by:

    1. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2017. "Record statistics of a strongly correlated time series: random walks and L\'evy flights," Papers 1702.00586, arXiv.org.
    2. Wergen, Gregor, 2014. "Modeling record-breaking stock prices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 396(C), pages 114-133.
    3. Srivastava, Shashi C.L. & Lakshminarayan, Arul, 2015. "Records in the classical and quantum standard map," Chaos, Solitons & Fractals, Elsevier, vol. 74(C), pages 67-78.
    4. Léo Régnier & Maxim Dolgushev & Olivier Bénichou, 2023. "Record ages of non-Markovian scale-invariant random walks," Nature Communications, Nature, vol. 14(1), pages 1-7, December.
    5. Claude Godreche & Satya N. Majumdar & Gregory Schehr, 2015. "Record statistics for random walk bridges," Papers 1505.06053, arXiv.org, revised Jan 2016.
    6. Gregory Schehr & Satya N. Majumdar, 2013. "Exact record and order statistics of random walks via first-passage ideas," Papers 1305.0639, arXiv.org.

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