Large deviations of the maximum of independent and identically distributed random variables

TL;DR

This paper explores the probability of unusually large maximum values in sets of i.i.d. random variables using Large Deviation Theory, providing detailed analysis for exponential and Gaussian cases and deriving a general rate function.

Contribution

It introduces a novel approach connecting Large Deviation Theory with Extreme Value Statistics, deriving a general rate function for the Gumbel class and highlighting additional information beyond the limiting distribution.

Findings

Rate function for maximum deviations derived for exponential and Gaussian variables

Full rate function contains more information than the Gumbel distribution alone

Provides detailed pedagogical derivation accessible to physicists

Abstract

A pedagogical account of some aspects of Extreme Value Statistics (EVS) is presented from the somewhat non-standard viewpoint of Large Deviation Theory. We address the following problem: given a set of $N$ i.i.d. random variables $\{X_1,\ldots,X_N\}$ drawn from a parent probability density function (pdf) $p(x)$, what is the probability that the maximum value of the set $X_{\mathrm{max}}=\max_i X_i$ is "atypically larger" than expected? The cases of exponential and Gaussian distributed variables are worked out in detail, and the right rate function for a general pdf in the Gumbel basin of attraction is derived. The Gaussian case convincingly demonstrates that the full rate function cannot be determined from the knowledge of the limiting distribution (Gumbel) alone, thus implying that it indeed carries additional information. Given the simplicity and richness of the result and its derivation, its absence from textbooks, tutorials and lecture notes on EVS for physicists appears inexplicable.