Extreme Value Theory: Gumbel, Fréchet & Weibull

Block maxima convergence to the GEV family. Draw samples from different parent distributions, compute block maxima, and watch the empirical distribution converge to its extreme value attractor.

GEV family: G(x; μ, σ, ξ) = exp{−[1 + ξ(x−μ)/σ]−1/ξ}
ξ = 0 (Gumbel): exponential-tailed parents (Normal, Exponential)
ξ > 0 (Fréchet): heavy-tailed parents (Pareto, log-Normal)
ξ < 0 (Weibull): bounded-tail parents (Uniform, Beta)
Fisher-Tippett-Gnedenko theorem guarantees convergence for large block size n.
Parent distribution
Block size n50
Number of blocks2000
Pareto tail α2.0
Empirical block maxima
Fitted GEV (MLE)
Theoretical attractor