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Statistics of Superior Records

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.