Density of Near-Extreme Events

We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential, faster than pure exponential, or as a pure exponential function. We argue that some of these results would remain valid even for certain correlated cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.

Figure 1

The lines closer to the dashed lines correspond to larger values of $N$. (a) $\overline{\rho (r,N)}$ for $N={10}^2$ (blue), ${10}^3$ (red), and ${10}^4$ (green), for the power-law distribution $p(X)=\alpha \exp (-X^{-\alpha })X^{-(1+\alpha )}$, with $\alpha =2$. The dashed (black) line plots the Fréchet distribution $f_1(r/b_N)$. (b) $\overline{\rho (r,N)}$ for exponential decay $p(X)=\delta X^{\delta -1}\exp (-X^{\delta })$. (i) For $\delta =1/2$, with $N={10}^3$ (blue), ${10}^5$ (red), and ${10}^7$ (green). The dashed (black) line plots the Gumbel distribution $f_2([r-a_N]/b_N)$. (ii) For $\delta =2$, with $N={10}^3$ (blue), ${10}^6$ (red), and ${10}^9$ (green). The dashed (black) line plots $p(a_N-r)$. (c) $\overline{\rho (r,N)}$ for bounded distribution, $p(X)=\beta a^{-\beta }(a-X{)}^{\beta -1}$ for $X<a$ and $p(X)=0$ for $X\ge a$, where $a=10$. (i) For $\beta =3/2$, with $N={10}^2$ (blue), ${10}^3$ (red), and ${10}^4$ (green). (ii) For $\beta =1/2$, with $N=10$ (blue), ${10}^2$ (red), and ${10}^3$ (green). The dashed (black) lines plot $p(a-r)$.

Figure 2

(a) Yamal peninsula June through July mean temperature anomaly ($\Delta T$) reconstruction series [15]. (b) The histogram plots the distribution $\Delta T$ of the data shown in (a). The solid line represents $p(\Delta T)=\exp (-\Delta T^2/2)/\sqrt{2\pi }$. In (c) and (d), the histograms plot the mean DOS relative to the maximum (excluding the maximum), computed by dividing the data into blocks, where each block consists of $N$ years. Solid lines are calculated using the exact numerical integration in Eq. (3). The dashed lines represent $p(a_N-r)$, where $a_N=(2\ln N{)}^{1/2}-(2\ln N{)}^{-1/2}(\ln \ln N+\ln 4\pi )/2$.