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 ρ. 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 ρ. We find that the fraction of superior sequences S(N) decays algebraically with sequence length N, S(N)~N(-β) in the limit N→∞. Interestingly, the decay exponent β is nontrivial, being the root of an integral equation. For example, when ρ is a uniform distribution with compact support, we find β=0.450265. In general, the tail of the parent distribution governs the exponent β. 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(-α). We use the above statistical measures to analyze earthquake data.