Mathematical Foundations for Fat Tails

Essential concepts in probability theory: random variables, distributions, and their properties.

• Random Variables and Distributions
• Expectation and Moments
• Key Properties of Expectation

The key probability distributions you need to know, from the well-behaved Gaussian to the wild Pareto.

• The Gaussian Distribution
• The Exponential Distribution
• The Pareto Distribution
• Student's t-Distribution
• Stable Distributions

Defining fat tails precisely, the subexponential class, and practical methods for detecting heavy tails in data.

• Defining Fat Tails
• The Subexponential Class
• Practical Detection

The Law of Large Numbers and Central Limit Theorem — when they work, when they fail, and what happens under fat tails.

• The Law of Large Numbers
• The Central Limit Theorem
• When CLT Fails
• Preasymptotics

Why standard statistics fail under fat tails and what to do instead.

• Why Standard Statistics Fail
• The Problem of Extremes
• Better Estimators
• Key Messages

The mathematics of extreme events — what distributions do maxima follow?

• The Setup
• The Three Types
• Domain of Attraction
• Implications for Risk

The conceptual framework that ties the mathematics together.

• Mediocristan vs Extremistan
• The Turkey Problem
• Convexity and Jensen's Inequality
• The Masquerade Problem
• Fragility and Antifragility

Optional but helpful mathematical foundations for the advanced reader.

• Why Measure Theory?
• Key Concepts
• Convergence Types
• Characteristic Functions

Test your understanding with carefully designed problems.

• Problem Set 1: Distributions
• Problem Set 2: Fat Tail Behavior
• Problem Set 3: Estimation
• Problem Set 4: Conceptual

Worked solutions with detailed explanations.

• Solutions and Intuition