Accurately approximating extreme value statistics

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N -> infinity, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with N is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.

Automated summary

This paper discusses the classic problem of extreme value statistics, showing that the distribution of maxima converges to one of three limiting forms through the Fisher-Tippett-Gnedenko theorem. Utilizing the Gumbel limit allows for accurate approximation of the extreme value distribution, with parameters represented as power series and the underlying distribution transformed. Functional corrections to the Gumbel limit are considered, obtainable through Taylor expansion, which also helps characterize extreme value statistics in cases where the underlying distribution is unknown.