E. Ben-Naim, P. L. Krapivsky
We study statistics of records in a sequence of random variables. The running record equals the maximum of all elements in the sequence up to a given point, and we define a superior sequence as one where all running records are above average. We obtain the record distribution for superior sequences in the limit N--> infinity where N is sequence length. Further, we find that the fraction of superior sequences S_N decays algebraically, S_N ~ N^{-beta}. Interestingly, the decay exponent beta is nontrivial, being the root an integral equation. For example, when the random variables are drawn from a uniform distribution, we find beta=0.450265. In general, the tail of the distribution function from which the random variables are drawn governs the exponent beta. We also consider the dual question of inferior sequences, where all records are below average, and find that the faction of inferior sequences I_N decays algebraically, albeit with a different decay exponent, I_N ~ N^{-alpha}.
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http://arxiv.org/abs/1305.4227
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