## Abstract

We consider the extreme value statistics of N independent and identically distributed random variables, which is a classic problem in probability theory. When N → ∞, 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.

Original language | English |
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Article number | 315205 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 54 |

Issue number | 31 |

DOIs | |

State | Published - Aug 2021 |

### Bibliographical note

Funding Information:The support of the Israel Science Foundation, Grant No. 1898/17, is acknowledged.

Publisher Copyright:

© 2021 IOP Publishing Ltd.

## Keywords

- Lambert scaling
- extreme value theory
- large deviations theory
- limiting distribution
- slow convergence