Outline for the extreme value theory lecture, March 13

I intend to cover the following topics: 

(1)    The Poisson distribution as the "law of rare events" (known, cf. TK).

(2)     A motivating analogy: The central limit theorem (CLT) for suitably scaled sums X₁ + ... + X_n, where {X_i} i.i.d. copies of X

• what distributions for X have the property that a linear combination a₁ X₁ + a_n X_n is distributed as c_n X + d_n (for suitable sequences c, d)?
• these are precisely the limiting distributions for the CLT. For finite-variance distributions: only the Gaussian. 
          This is *the* reason why the Gaussian shows up "everywhere".

(3)    Instead of sum{X_i}, what about maxima M_n = max{X₁ + ... + X_n} (suitably scaled)? How are they distributed, asymptotically?

• limit distributions are precisely the same as the ones for which M_n is distributed c_n X + d_n : Fréchet, Gumbel and Weibull
• all three in unified parametrization: the "generalized extreme value distribution" (GEV).

(4)    So scaled M_n → GEV in distribution, but which GEV?

• Power tails (1 - cdf ~ x^a) to Fréchet
• Lighter (unbounded) tails to Gumbel
• Bounded tails to Weibull (in terms of transformation to Fréchet).

(5)    Other characterizations, and connection to "excess over threshold"

• The generalized Pareto distribution (GPD)
•         A few properties.

(6)    Bringing it all together:

• For a compound Poisson process of exceedences with GPD distributions, the max is GEV distributed
• In practice: for sufficiently rare events (i.e. sufficiently high thresholds), we have approximate Poisson arrivels of approximate GPD exceedences, and could hope for approximate GEV maxima
• What is "sufficiently high" threshold? Where mean exceedence as function of threshold, is "sufficiently close to" linear.

(7)    If time permits: Inference from data.