Stable Distributions — Mathematics Knowledge System

Overview

A probability distribution is stable (or α-stable) if a linear combination of two independent copies of a variable with that distribution has the same distribution (up to location and scale). The normal and Cauchy distributions are the best-known special cases. Stable distributions are the only possible limiting distributions for normalised sums of i.i.d. variables (the generalised central limit theorem), making them fundamental in probability theory. They arise naturally in financial modelling, signal processing, and physics whenever one encounters heavy-tailed phenomena that violate the finite-variance assumption of the classical CLT.

Intuition

The classical CLT tells us that sums of i.i.d. variables with finite variance converge (after normalisation) to a normal distribution. But what if the variance is infinite — as happens with power-law distributions? The generalised CLT says sums still converge, but to an α-stable distribution with α < 2 instead of the normal. The parameter α controls the tail heaviness: smaller α means heavier tails. The Cauchy distribution (α = 1) is so heavy-tailed that even the mean is undefined. Stable distributions are the 'attractors' in the space of distributions, just as the normal distribution is the attractor for finite-variance distributions.

Formal Definition

Definition

A random variable X is said to have a stable distribution if for independent copies X₁ and X₂, there exist constants c > 0 and d ∈ ℝ such that X₁ + X₂ =ᵈ cX + d. The characteristic function of a stable distribution with stability index α ∈ (0,2] is:

\log \varphi_X(t) = i\mu t - |\sigma t|^{\alpha}\!\left(1 + i\beta\,\operatorname{sgn}(t)\,\omega(t,\alpha)\right)

Log-characteristic function of a stable distribution

\omega(t,\alpha) = \begin{cases} \tan(\pi\alpha/2) & \alpha \neq 1 \\ \frac{2}{\pi}\log|t| & \alpha = 1 \end{cases}

Auxiliary function ω

\alpha = 2 \implies X \sim N(\mu, 2\sigma^2) \quad (\text{normal})

Special case α = 2: normal distribution

\alpha = 1,\, \beta = 0 \implies X \sim \operatorname{Cauchy}(\mu, \sigma) \quad (\text{Cauchy})

Special case α = 1, β = 0: Cauchy distribution

Worked Examples

1
The characteristic function of Cauchy(0,1) is φ(t) = e^{−|t|}.
$\varphi_{X_1}(t) = \varphi_{X_2}(t) = e^{-|t|}$
2
For independent variables, characteristic functions multiply.
$\varphi_{X_1+X_2}(t) = e^{-|t|} \cdot e^{-|t|} = e^{-2|t|}$
3
Recognise e^{−2|t|} as the characteristic function of Cauchy(0,2).
$e^{-2|t|} = \varphi_{\operatorname{Cauchy}(0,2)}(t)$

✓ Answer

X₁ + X₂ ~ Cauchy(0,2) — the Cauchy distribution is stable with index α = 1.

Practice Problems

Mediumfree response

What does the stability index α control in a stable distribution?

Hardfree response

State the generalised central limit theorem involving stable distributions.

Quiz

Summary

A stable distribution has stability index α ∈ (0,2]: the sum of independent copies is stable with the same α.
Special cases: α = 2 is the normal distribution; α = 1 with β = 0 is the Cauchy distribution.
For α < 2, tails are power-law (heavier than normal), and variance is infinite; for α ≤ 1, the mean is also undefined.
Stable distributions are the limiting distributions in the generalised central limit theorem for heavy-tailed i.i.d. variables.

References

WebsiteWikipedia — Stable distribution