Renormalization flow for extreme value statistics of random variables raised to a varying power

TL;DR

This paper investigates the asymptotic behavior of the maximum of i.i.d. random variables raised to a varying power, revealing a transition between classical and non-standard limit distributions through a renormalization approach.

Contribution

It introduces a renormalization framework to analyze the limit distributions of maxima of powered variables with a varying exponent, uncovering a transition mechanism between classical and non-standard distributions.

Findings

Standard limit distributions are recovered when q(n) grows slower than q*(n).

Non-standard limit distributions emerge when q(n) behaves asymptotically as k.q*(n).

A transition mechanism between classical and non-standard distributions is identified.

Abstract

Using a renormalization approach, we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power q(n) that varies monotonically with the sample size n. Under these conditions, a non-standard class of max-stable limit distributions, which mirror the classical ones, emerges. Furthermore a transition mechanism between the classical and the non-standard limit distributions is brought to light. If q(n) grows slower than a characteristic function q*(n), the standard limit distributions are recovered, while if q(n) behaves asymptotically as k.q*(n), non-standard limit distributions emerge.