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

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 lambda q(n)*, nonstandard limit distributions emerge.