# 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 X_{1} + ... + X_{n}, where {X_{i}} i.i.d. copies of X

- what distributions for X have the property that a linear combination a
_{1}X_{1}+ 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{X_{1} + ... + 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.