Domain of Attraction

Which limit applies? How tail behavior determines the extreme value type.

Given a distribution , which of the three extreme value types does it belong to? The answer depends on the tail behavior of . This section develops the mathematical criteria that determine the domain of attraction.

What is a Domain of Attraction?

Definition

Domain of Attraction

A distribution is in the domain of attraction of an extreme value distribution (written ) if there exist normalizing constants and such that:

for all where is continuous.

The domain of attraction tells us what type of extremes to expect. The classification depends entirely on how the survival function behaves as .

Slowly Varying Functions: The Key Building Block

Before diving into the domain classifications, we need to understand a fundamental concept that appears throughout extreme value theory: slowly varying functions.

Key Insight

The Intuition

Pure power laws like are mathematically clean but rarely perfect in practice. Real distributions often deviate slightly from pure power laws while still being "essentially" power-law in their tail behavior.

Slowly varying functions capture these small deviations without changing the fundamental tail classification.

Definition

Slowly Varying Function

A function is slowly varying at infinity if:

In words: scaling the input by any constant eventually has negligible effect on the function's value.

Examples of Slowly Varying Functions

Function L(x)	L(2x)/L(x) at x = 10	L(2x)/L(x) at x = 100	Limit
c (constant)	1.00	1.00	1
ln(x)	1.301	1.151	1
ln(ln(x))	1.302	1.150	1
(ln x)²	1.693	1.326	1

Notice how the ratios get closer to 1 as x increases. This is the hallmark of slow variation.

Why "Slowly"?

Slowly varying functions grow slower than any positive power of :

Example

Comparing Growth Rates

At x = 1,000,000:

Wait, is bigger! But as x continues to grow, eventually dominates. At :

Still ln wins! But at :

(a googol!)

Any power eventually wins, no matter how small.

Interactive: Slowly Varying Functions

Slowly varying functions satisfy L(tx)/L(x) → 1 as x → ∞ for any constant t > 0

Choose L(x):

tscaling factor

2.00

L(x) and L(2x)

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Both functions grow, but the gap between them stabilizes

Ratio L(2x) / L(x)

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Ratio approaches 1 as x increases

Function:

L(x) = ln(x)

Ratio at x = 1000:

L(2×1000) / L(1000) = 1.1003

Compare to Power Functions

For a power function g(x) = x^α, the ratio is:

g(tx) / g(x) = (tx)^α / x^α = t^α

This stays constant at t^α — it doesn't approach 1. That's the key difference: slowly varying functions "don't care" about multiplicative scaling.

The Frechet Domain: Regularly Varying Tails

Definition

Regular Variation

A function is regularly varying with index (written ) if:

for all .

Equivalently, can be written as:

where is a slowly varying function — a function such that as .

Definition

Slowly Varying Function

A function is slowly varying if:

for all . Examples include constants, ,, etc.

Key Insight

The Frechet Criterion

A distribution is in the Frechet domain if and only if its survival function is regularly varying:

The exponent α becomes the shape parameter of the limiting Frechet distribution. This is precisely the class of fat-tailed distributions.

📚

Prerequisite

Section 3.1: Defining Fat Tails

For the formal hierarchy of thin to fat-tailed distributions.

→

Example

Pareto Distribution

For Pareto(, ):

Here is constant — the simplest slowly varying function. So Pareto is in the Frechet domain with parameter .

Example

Student's t Distribution

For Student's t with ν degrees of freedom:

This is regularly varying with index , so t-distributions are in the Frechet domain with .

The Gumbel Domain: Exponential-Type Tails

The Gumbel domain includes distributions whose tails decay faster than any power law but not as fast as being bounded.

Definition

Gumbel Domain Criterion

A distribution is in the Gumbel domain if there exists a positive function such that:

where is the right endpoint of (possibly infinite).

Intuitively, distributions in the Gumbel domain have tails that decay exponentially or faster (but are unbounded):

Gaussian: — faster than exponential
Exponential: — exactly exponential
Gamma: — exponential times polynomial

Example

Exponential Distribution

For the Exponential() distribution:

With , we have:

So Exponential is in the Gumbel domain.

Key Insight

Gumbel = Light Tails

If your distribution is in the Gumbel domain, you're dealing with light tails. Extremes are relatively predictable, the maximum grows slowly with sample size, and traditional statistical methods are more reliable.

The Weibull Domain: Bounded Distributions

Definition

Weibull Domain Criterion

A distribution with finite right endpoint is in the Weibull domain if:

for some and slowly varying .

Example

Uniform Distribution

For Uniform(0, 1), and for . Thus:

So Uniform(0, 1) is in the Weibull domain with .

Interactive Domain Classifier

Use this tool to explore how different distributions are classified into their domains of attraction. Select a distribution and adjust its parameters to see the shape parameter ξ and understand the tail behavior.

Distribution:

tail exponent (α):2.0

🔴 Fréchet Domain

ξ = 0.50

Pareto has regularly varying tails: S(x) ~ x^{-α}. The tail exponent α = 2.0 gives ξ = 1/α = 0.50.

Maximum growth: M_n ~ n^{0.50}

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Reading the Log-Log Plot:

Fréchet: Linear with negative slope = -α (power law tails)
Gumbel: Curves downward sharply (exponential or faster decay)
Weibull: Drops to -∞ at finite x (bounded distribution)

Domain	ξ value	Tail behavior	Risk implications
Fréchet	ξ > 0	Power law S(x) ~ x^-α	High tail risk, use EVT methods
Gumbel	ξ = 0	Exponential-type decay	Moderate tail risk
Weibull	ξ < 0	Bounded upper tail	Low tail risk, maximum is bounded

Practical Identification

How do you determine which domain your data belongs to?

Method 1: Log-Log Plot

Plot against . If the plot is:

Linear with slope : Frechet domain (fat tails)
Curves down sharply: Gumbel domain (light tails)
Terminates at finite x: Weibull domain (bounded)

Method 2: Mean Excess Function

The mean excess function behaves characteristically:

Increasing linearly: Frechet domain
Constant: Exponential (boundary of Gumbel)
Decreasing: Gumbel or Weibull domain

Method 3: Fit the GEV

Fit the GEV distribution to the block maxima of your data. The estimated shape parameter tells you:

(significantly): Frechet domain
: Gumbel domain
(significantly): Weibull domain

Key Insight

The Key Diagnostic

The most important question in EVT is: Is ? If yes, you're in fat-tail territory and must use fat-tail methods. If no, traditional approaches may work.

For financial returns, insurance claims, and natural disasters, the answer is almost always yes.

Summary: Domain Classification

Domain	Tail Condition	Examples	Max Growth
Frechet		Pareto, t, Cauchy
Gumbel	Exponential decay	Gaussian, Exp, Gamma	or
Weibull	Bounded support	Uniform, Beta	Approaches

Key Takeaways

A distribution's domain of attraction is determined by its tail behavior
Frechet domain: Regularly varying tails — this is the fat-tail domain
Gumbel domain: Exponential or faster decay — light tails
Weibull domain: Bounded distributions
Slowly varying functions capture deviations from pure power laws while preserving the essential tail behavior
Practical tools (log-log plots, mean excess, GEV fitting) can identify the domain from data