Statistics of Superior Records
E. Ben-Naim and 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 {\em superior}
sequence as one where all running records are above average. We
obtain the record distribution for superior sequences in the limit
$N\to \infty$ where $N$ is sequence length. Further, we find that
the fraction of superior sequences $S_N$ decays algebraically,
$S_N\sim N^{-\beta}$. 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, $I_N\sim N^{-\alpha}$.
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