Scaling Exponent for Incremental Records

Abstract

We investigate records in a growing sequence of identical and independently distributed random variables. The record equals the largest value in the sequence, and our focus is on the increment, defined as the difference between two successive records. We investigate sequences in which all increments decrease monotonically, and find that the fraction I_N of sequences that exhibit this property decays algebraically with sequence length N, namely I_N ~ N^{-nu} as N --> infinity. We analyze the case where the random variables are drawn from a uniform distribution with compact support, and obtain the exponent nu = 0.317621... using analytic methods. We also study the record distribution and the increment distribution. Whereas the former is a narrow distribution with an exponential tail, the latter is broad and has a power-law tail characterized by the exponent nu. Empirical analysis of records in the sequence of waiting times between successive earthquakes is consistent with the theoretical results.

7 pages, 8 figures

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Cited by in corpus (12)

• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Universal statistics of longest lasting records of random walks and Lévy flights
• Exact statistics of record increments of random walks and Lévy flights
• Record breaking bursts during the compressive failure of porous materials
• Record statistics for random walk bridges
• Record statistics of bursts signals the onset of acceleration towards failure
• Striking universalities in stochastic resetting processes
• Slow Kinetics of Brownian Maxima
• Persistence of Random Walk Records
• Exact solution for a sample space reducing stochastic process
• On the first positive position of a random walker
• Scaling Exponents for Ordered Maxima