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, $I_N\sim N^{-\alpha}$.


source, pdf