The Setup

What is the distribution of the maximum of n observations?

When dealing with risk, we often care most about the extremes. What's the worst flood in 100 years? The largest insurance claim? The maximum daily loss in a portfolio? Extreme Value Theory (EVT) provides the mathematical framework for understanding these questions.

EVT is one of the most powerful tools for fat-tailed analysis because it focuses exactly where fat tails matter most: in the extremes.

The Central Question

Suppose we observe independent and identically distributed (i.i.d.) random variables . A natural question arises:

Definition

Maximum of n Observations

Given i.i.d. observations, the maximum is:

What is the distribution of ? How does it behave as ?

Read: “M sub n equals the maximum of X one through X n”

The largest value among n observations

Why This Matters

Understanding the distribution of extremes is crucial in many fields:

  • Hydrology: What's the 100-year flood level for dam design?
  • Insurance: What's the largest claim we might face?
  • Finance: What's the worst-case portfolio loss?
  • Engineering: What loads must a structure withstand?
  • Climate: How extreme will temperature records get?
Key Insight

The Extreme Value Miracle

Just as the Central Limit Theorem tells us that sums of random variables tend toward the Gaussian distribution (under certain conditions), Extreme Value Theory tells us that maxima of random variables can only converge to one of three possible distributions — regardless of the original distribution! This is the Fisher-Tippett-Gnedenko theorem.

The CDF of the Maximum

If is the CDF of each , we can derive the CDF of the maximum:

Definition

CDF of the Maximum

The cumulative distribution function of is:

Since the observations are independent:

Example

Maximum of Uniform Random Variables

Let , so for . Then:

As grows, concentrates near 1. For example, with , .

The Need for Normalization

The raw maximum typically drifts to infinity (or to some upper bound) as . To get a meaningful limit, we need to normalize it:

where and are sequences of normalizing constants that depend on . The key question becomes:

For what distributions can we find constants and such that:

where denotes convergence in distribution.

Key Insight

The Remarkable Answer

The Fisher-Tippett-Gnedenko theorem gives a surprising answer: there are only three possible non-degenerate limiting distributions. We explore these in the next section.

Interactive: Block Maxima Method

Explore how block maxima from different parent distributions converge to different members of the GEV family:

Parent distribution:
mblock size
50
k# blocks
200
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Domain of Attraction

Gumbel (ξ = 0)

Shape Parameter

ξ ≈ 0.00

Gumbel domain (light tails): With Gaussian parent distribution, maxima follow a GEV with ξ = 0 (Gumbel distribution). The maximum grows slowly with sample size — approximately as log(n).

Block Maxima Method: Divide data into blocks of size m, take the maximum of each block. By the Fisher-Tippett-Gnedenko theorem, these maxima follow a GEV distribution:

  • ξ > 0 (Fréchet): Heavy-tailed parent — unbounded extremes
  • ξ = 0 (Gumbel): Light-tailed parent — bounded exponential tails
  • ξ < 0 (Weibull): Bounded parent — finite upper endpoint

Connection to Fat Tails

The behavior of extremes reveals the nature of a distribution's tails:

  • Thin-tailed distributions (like Gaussian): The maximum grows slowly, roughly as . Extremes are tightly controlled.
  • Fat-tailed distributions (like Pareto): The maximum grows as a power of , specifically as . Extremes can be enormous.
Example

Maximum Growth Rates

For observations:

  • Gaussian: Maximum is typically around 3.1 standard deviations above the mean
  • Pareto with : Maximum is typically about 31 times the minimum value (since )

The fat-tailed maximum is orders of magnitude larger relative to typical values!

Key Takeaways

  • Extreme Value Theory studies the distribution of maxima (and minima) of random samples
  • The CDF of the maximum is — this follows directly from independence
  • To get a non-trivial limit, we must normalize the maximum with sequences and
  • There are only three possible limiting distributions for normalized maxima — this is the Fisher-Tippett-Gnedenko theorem
  • The behavior of extremes differs dramatically between thin-tailed and fat-tailed distributions