Stable Distributions

The broader family of distributions that are "stable" under addition — including the Gaussian as a special case.

Stable distributions (also called α-stable or Lévy stable) form a special family: they are “stable” under addition, meaning that a sum of i.i.d. stable random variables has the same distribution type. The Gaussian is the most famous member, but it's the only stable distribution with finite variance.

The Stability Property

Before diving into the formulas, let's understand what makes stable distributions special: their behavior under addition.

Definition

Stability Under Addition

A distribution is stable if adding independent copies produces the same type of distribution (just rescaled). Formally, if and are independent copies:

Read: “X₁ plus X₂ has the same distribution as c times X plus d”

The sum looks like a scaled and shifted version of a single copy

Most distributions don't have this property. When you add two copies of a typical distribution, you get something different. But stable distributions maintain their shape.

Example

Why Gaussian is Stable

If are independent:

The sum is still Gaussian — just scaled by . This is the stability property in action. For stable distributions with index α, the scale factor is .

Explore: Stability Under Addition

See how adding copies of a stable distribution gives the same shape back. Adjust α to see how the stability index controls both tail thickness and the scaling behavior.

What “Stability” Means

A distribution is stable if adding copies gives the same distribution type, just rescaled. Watch how the sum of 2 copies (orange) matches the original (blue) after proper rescaling.

αstability

1.50

n# of copies

The stability property says:

X₁ + X₂ ~ 1.59 × X

Scale factor: n^1/α = 2^1/1.5 = 1.59

Loading chart...

The dashed line (sum of 2 copies, properly rescaled) overlays the solid line — same shape!

The α Parameter: Tail Thickness

α = 2: Gaussian

Thinnest tails possible for a stable distribution. All moments exist. Variance is finite.

α = 1: Cauchy

No mean exists! Tails decay as x⁻². The sample mean never converges.

α < 1: Extreme

Even fatter tails. Dominated by rare, massive events. The Lévy distribution (α = 0.5) is an example.

Taleb's key insight about stable distributions:

Gaussian (α = 2) is the only stable distribution with finite variance
All other stable distributions (α < 2) have infinite variance
There's no middle ground — you're either Gaussian or fat-tailed
If your data is stable but not Gaussian, standard statistics (based on variance) fail

Key Insight

Why Stability Matters

The Generalized Central Limit Theorem says: if sums of i.i.d. random variables converge to anything, they must converge to a stable distribution. This means:

Finite variance → sums converge to Gaussian (α = 2)
Infinite variance → sums converge to non-Gaussian stable (α < 2)
Stable distributions are the only possible limit laws

What Does “Stable” Mean?

Definition

Stable Distribution

A distribution is stable if for any i.i.d. copies and , there exist constants a > 0 and b such that:

Read: “X₁ plus X₂ has the same distribution as aX plus b”

The sum has the same type of distribution as a single observation, just rescaled and shifted

This is a remarkable property. For most distributions, adding two copies creates something different. But stable distributions maintain their character under addition.

Example

Gaussian Stability

If are independent, then:

The sum is still Gaussian, with doubled mean and variance. This is why the Gaussian is stable (and is sometimes called “2-stable”).

The Four Parameters

A general stable distribution is characterized by four parameters:

α (stability index): 0 < α ≤ 2. Controls tail thickness.
- α = 2: Gaussian (thinnest tails among stable)
- α = 1: Cauchy (no mean)
- Smaller α = fatter tails
β (skewness): -1 ≤ β ≤ 1. Controls asymmetry.
- β = 0: Symmetric
- β = 1: Maximally right-skewed
- β = -1: Maximally left-skewed
γ (scale): γ > 0. Controls spread (like σ for Gaussian).
δ (location): Any real number. Shifts the distribution.

We write or simply when parameters are understood.

Key Special Cases

α = 2: Gaussian

The only stable distribution with finite variance. Symmetric (β = 0 effectively). Familiar bell curve behavior.

α = 1, β = 0: Cauchy

Symmetric but with no mean. Tails decay as . The ratio of two independent standard normals follows Cauchy.

α = 0.5, β = 1: Lévy

Extremely fat-tailed, defined only for x > δ. The hitting time of Brownian motion follows a Lévy distribution.

The Characteristic Function

Most stable distributions don't have closed-form PDFs. They're defined through their characteristic function:

where:

Key Insight

Why Characteristic Functions?

Characteristic functions always exist (unlike moments) and uniquely determine the distribution. They make stability easy to verify: X is stable iff for some .

🎓

For rigorous treatment

Section 8.4: Characteristic Functions

For a rigorous treatment of characteristic functions and why they always exist.

→

Moment Structure

For stable distributions with stability index α:

α = 2 (Gaussian): All moments exist
α < 2: Variance is infinite (doesn't exist)
α ≤ 1: Mean doesn't exist either
General rule: only if

Key Insight

Taleb's Point About Variance

The Gaussian is the only stable distribution with finite variance. All other stable distributions have infinite variance. This means stable distributions are either:

Gaussian (α = 2): well-behaved with finite variance
All others (α < 2): fat-tailed with infinite variance

There's no “middle ground” — stability forces you to either Gaussian or fat tails.

Generalized Central Limit Theorem

Why do stable distributions matter? They're the only possible limits for normalized sums:

Definition

Generalized CLT

If are i.i.d. and there exist normalizing constants , such that:

then Y must be a stable distribution.

The classical CLT says: if variance is finite, the limit is Gaussian. But if variance is infinite (fat tails), and a limit exists, it must be a non-Gaussian stable distribution.

Example

Stable Limit for Fat Tails

Sum of i.i.d. Pareto(α) with 1 < α < 2:

Mean exists, variance is infinite
Properly normalized sums converge to an α-stable distribution
Not Gaussian! The limit has the same tail exponent α

Explore: The Stable Distribution Family

Adjust α to see how the distribution changes from Gaussian (α=2) through Cauchy (α=1) to extremely fat-tailed (small α). Observe how β controls skewness.

αstability index

1.50

βskewness

0.00

Compare to Gaussian

S_1.5(β=0.0)

Mean exists, but variance is infinite (α = 1.5 < 2)

Probability Density Function (Approximate)

Loading chart...

Note: This visualization uses a Student-t approximation to illustrate stable distribution shapes. True stable PDFs (except Gaussian and Cauchy) don't have closed-form expressions.

Taleb's Key Point: Gaussian Uniqueness

Among all stable distributions, the Gaussian (α = 2) is the only one with finite variance. All other stable distributions (α < 2) have infinite variance.

α = 2 (Gaussian)

All moments exist
Variance = σ² (finite)
CLT applies normally

α < 2

Variance = ∞
Mean exists only if α > 1
Sums stay stable (same α)

What “Stable” Means

If X₁ and X₂ are independent copies of an α-stable distribution:

X₁ + X₂ = c · X + d

for some constants c and d. The sum has the same type of distribution (just rescaled). For Gaussian: c = √2, d = 0, giving X₁ + X₂ ~ N(0, 2σ²).

This means: sums of fat-tailed (α < 2) stable variables remain fat-tailed. They don't “Gaussianize” — the Generalized CLT says they converge to the same α-stable distribution.

Notable Stable Distributions

α = 2, β = 0Gaussian (Normal)

α = 1, β = 0Cauchy

α = 0.5, β = 1Lévy

Practical Implications

Modeling fat-tailed phenomena: Stable distributions can model data that the Gaussian can't, like financial returns or insurance claims.
Aggregation: Sums of fat-tailed data don't “Gaussianize” — they stay fat-tailed with the same α.
Risk estimation: Standard methods (based on variance) fail for α < 2 stable distributions.
Simulation: You can generate stable random variables (Chambers-Mallows-Stuck algorithm), enabling Monte Carlo studies.

Example

Mandelbrot and Cotton Prices

Benoit Mandelbrot (1963) proposed modeling cotton price returns with stable distributions (α ≈ 1.7). This was one of the first serious challenges to Gaussian finance, predating Taleb's work by decades.

Key Takeaways

Stable distributions are preserved under addition (up to rescaling)
Four parameters: α (stability), β (skewness), γ (scale), δ (location)
α = 2 is Gaussian; α = 1 is Cauchy; smaller α means fatter tails
Gaussian is the only stable distribution with finite variance
The Generalized CLT: sums of i.i.d. variables can only converge to stable distributions
For fat-tailed data (α < 2), sums stay fat-tailed — they don't become Gaussian