Statistics of Superior Records

Abstract

We study statistics of records in a sequence of random variables. These identical and independently distributed variables are drawn from the parent distribution rho. The running record equals the maximum of all elements in the sequence up to a given point. We define a superior sequence as one where all running records are above the average record, expected for the parent distribution rho. We find that the fraction of superior sequences S_N decays algebraically with sequence length N, S_N ~ N^{-beta} in the limit N-->infty. Interestingly, the decay exponent beta is nontrivial, being the root of an integral equation. For example, when rho is a uniform distribution with compact support, we find beta=0.450265. In general, the tail of the parent distribution governs the exponent beta. We also consider the dual problem of inferior sequences, where all records are below average, and find that the fraction of inferior sequences I_N decays algebraically, albeit with a different decay exponent, I_N ~ N^{-alpha}. We use the above statistical measures to analyze earthquake data.

8 pages, 6 figures, expanded version

References in corpus (6)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Record statistics for biased random walks, with an application to financial data
• Record Statistics for Multiple Random Walks
• Nonlinear theory and tests of earthquake recurrence times
• Are megaquakes clustered?
• Recurrence Statistics of Great Earthquakes

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 statistics of financial time series and geometric random walks
• Record statistics for random walk bridges
• Scaling Exponent for Incremental Records
• Slow Kinetics of Brownian Maxima
• Record Ages of Scale Invariant non-Markovian Random Walks
• Persistence of Random Walk Records
• On the first positive position of a random walker
• Scaling Exponents for Ordered Maxima
• Hiring Strategies