The Subexponential Class
The mathematical property that defines fat tails — when the maximum dominates the sum.
The subexponential class provides the mathematical foundation for understanding fat tails. Its defining property — that the sum is dominated by the maximum — captures exactly why fat-tailed phenomena behave so differently from what our intuition (trained on thin tails) expects.
The Defining Property
Subexponential Distribution
A distribution F with support on [0, ∞) is subexponentialif for independent X₁, X₂ with distribution F:
Read: “The limit as x approaches infinity of P(X₁ + X₂ exceeds x) over P(X₁ exceeds x) equals 2”
The probability that the sum exceeds x is asymptotically twice the probability that one term exceeds x
Why “2”? Because the sum exceeds x primarily through one of two mutually exclusive events: either X₁ > x OR X₂ > x. The probability that both are large is negligible compared to one being large.
Contrast with Light Tails
For light-tailed distributions (like Gaussian), the sum exceeds x through a collective contribution of many moderate values:
Light Tails (Gaussian)
If X₁ + X₂ = 100, typical scenario:
- X₁ ≈ 50, X₂ ≈ 50
- Both contribute moderately
- No single term dominates
Fat Tails (Pareto)
If X₁ + X₂ = 100, typical scenario:
- X₁ ≈ 98, X₂ ≈ 2
- Or: X₁ ≈ 3, X₂ ≈ 97
- One term dominates
The Catastrophe Principle
The subexponential property extends to sums of any size:
More strikingly, this is equivalent to:
Catastrophe Principle
For subexponential distributions:
Read: “The probability that the sum exceeds x is asymptotically equal to the probability that the maximum exceeds x”
The sum is big when and only when the maximum is big — one term causes the 'catastrophe'
The Single Event Dominance
In fat-tailed systems, large outcomes come from single extreme events, not accumulation of many moderate events. This is why:
- One earthquake causes more damage than 100 smaller ones
- One book sale (Harry Potter) dwarfs thousands of average books
- One trading loss can exceed all previous profits
Explore: Sum vs Maximum
Sample from different distributions and observe how the sum relates to the maximum. In fat-tailed distributions, they track closely.
Properties of Subexponential Distributions
Closure Properties
- Products: If X is subexponential, so is cX for any c > 0
- Mixtures: Mixtures of subexponential distributions are subexponential
- Convolutions: The sum of independent subexponentials with the same tail index is subexponential
Important Examples
| Distribution | Tail Behavior | Subexponential? |
|---|---|---|
| Pareto | Yes | |
| Log-normal | Yes (borderline) | |
| Weibull (β < 1) | Yes | |
| Exponential | No (boundary) | |
| Gaussian | No |
Implications for Practice
Insurance and Risk
Consider an insurance company with 1000 policies, each with independent claim sizes.
If claims are Gaussian:
- Total claims ≈ sum of many similar-sized claims
- Easy to predict with Law of Large Numbers
- Diversification works perfectly
If claims are Pareto:
- Total claims ≈ one massive claim
- Historical averages are misleading
- Diversification provides limited protection
Investment Returns
Annual returns on an aggressive strategy over 20 years:
If returns are Gaussian:
- Cumulative return = steady accumulation
- Each year contributes roughly equally
If returns are fat-tailed:
- A few exceptional years dominate your wealth
- Missing the best 5 days can destroy decades of returns
- Similarly, one crash can wipe out decades of gains
Why Diversification Has Limits
In Gaussian world, adding more independent assets reduces risk as 1/√n. In fat-tailed world, diversification helps less because single events dominate. You can't diversify away the maximum.
Key Takeaways
- Subexponential =
- The sum exceeds x when ONE term exceeds x, not through accumulation
- Catastrophe principle:
- Pareto, log-normal, and Weibull (β < 1) are subexponential
- In subexponential domains, single events dominate — diversification has limits