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 $N$, $S_N\sim N^{-\beta }$ in the limit $N\to \infty$. 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, $I_N\sim N^{-\alpha }$. We use the above statistical measures to analyze earthquake data.

Figure 1

The fraction $S_N$ of sequences with superior records versus the number of random variables $N$. The random variables are drawn from the uniform distribution (5). The results represent an average over ${10}^8$ independent Monte Carlo realizations.

Figure 2

The scaling behavior (11). Shown are the normalized polynomials ${\Phi }_N=F_N/S_N$ versus the variable $s$ for $N\le 4$. Also shown for reference is the scaling function (14).

Figure 3

The scaling functions $\Phi \left(s\right)$ and $\varphi \left(s\right)={\Phi }^{\prime }\left(s\right)$ versus the scaling variable $s$ for $\alpha =1$.

Figure 4

The exponents $\alpha$ (top panel) and $\beta$ (bottom panel) versus the parameter $\mu$. The dashed lines indicate the lower bound, ${\alpha }_{\min }=0.561459$, and upper bound, ${\beta }_{\max }=0.621127$, respectively.

Figure 5

The average record versus sequence length. Shown are empirical results for the earthquakes with magnitude $M>7$ and for earthquakes with magnitude $M>5$. Also shown for a reference is the harmonic number (29) that corresponds to the Poisson process.

Figure 6

The probabilities $S_N$ and $I_N$ for the earthquake sequence with $M>7$ (bottom panel) and $M>5$ (top panel). The empirical results for $M>7$ are compared with the theoretical predictions (10) and (39) with $R_N=\exp (-A_N)$ with $A_N$ given in (29).