Universal Record Statistics of Random Walks and Lévy Flights

Abstract

It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution grows as the sqrt(4N/pi) while the standard deviation grows as sqrt((2-4/pi) N), so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt(N/pi) and 0.626508... N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.

4 pages, 3 figures. Added journal ref. and made small changes. Compatible with published version

References in corpus (11)

• On the Role of Global Warming on the Statistics of Record-Breaking Temperatures
• Evolution in complex systems
• Breaking records in the evolutionary race
• Records in a changing world
• Scaling in Tournaments
• Evolutionary trajectories in rugged fitness landscapes
• ``Weather'' Records: Musings on Cold Days after a Long Hot Indian Summer
• Punctuated evolution for the quasispecies model
• Universal Extremal Statistics in a Freely Expanding Jepsen Gas
• Exact solution of a model of time-dependent evolutionary dynamics in a rugged fitness landscape
• Rank Statistics in Biological Evolution

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