Explain distribution of maxima & minima-Extreme Value Theory

Question

Illustrates an example of distribution of maxima and minima in Extreme Value Theory?

Answer 1

One significant EVT result concerns the distribution of maxima and minima and is used in computations as in the example given.

If Xi are independent, evenly distributed random variables and x = max(X1, X2, ... , Xn) therefore the distribution of x converges to

(1/σ) (1 + ξ (ξ(x -µ)/ σ))(-1/ ξ)-1 exp(-(1 + ξ(x - µ)/ σ))(-1/ ξ))

While ξ = 0 it is a Gumbel distribution, ξ< 0 that is Weibull and ξ >0 that is Frechet. Here Frechet is the one of interest in finance because this is related with fat tails. The role of theorems regarding extremes is similar to which of the Central Limit Theorem for sums/averages.