Density of near-extreme events

TL;DR

This paper analyzes the density of states near the maximum of i.i.d. random variables, revealing three limiting forms depending on tail decay, with implications for correlated cases and real-world temperature data.

Contribution

It provides an exact computation of the density of states near the maximum and explores its limiting behaviors for different tail decays, extending to correlated variables.

Findings

Mean DOS converges to three different forms based on tail decay.

Results are applicable to certain correlated variables, verified with Gaussian sequences.

Temperature data shows crowding consistent with theoretical predictions.

Abstract

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, faster than, or as a pure exponential function. We argue that some of these results would remain valid even for certain {\em 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.