08. Alpha-Stable Distributions — The Universal Law of Heavy Tails

Distribution 3/3. Alpha-stable (α-stable, Lévy stable) distributions are a four-parameter family defined by the stability axiom that "their shape does not change under independent sums." They include Gaussian and Cauchy distributions as special cases and provide the CLT limits for infinite-variance data. Background: characteristic functions, stability and the generalized CLT.

Definition: the stability axiom

A random variable $X$ is stable if, for independent copies $X_1, X_2$ and arbitrary $a, b > 0$, there exist some $c > 0, d \in \mathbb{R}$ such that

$$aX_1 + bX_2 \overset{d}{=} cX + d$$

There must then exist an exponent $\alpha \in (0, 2]$ for which $c$ satisfies $c^{\alpha} = a^{\alpha} + b^{\alpha}$. This $\alpha$ is the stability index, and it governs the character of the distribution.

• $\alpha = 2$: $c^2 = a^2 + b^2$—additivity of variance, hence the Gaussian
• $\alpha \lt 2$: the world of infinite variance and heavy tails

As guaranteed by the generalized CLT, stable distributions are all the possible limits of i.i.d. sums. If the Gaussian is "the universal limit in the finite-variance world," the alpha-stable family is the complete list of universal limits after removing that condition.

Defining them with characteristic functions

Except for three special cases, alpha-stable distributions have no closed-form density. Their characteristic functions, however, always have a closed form. In the standard parameterization (1-parameterization) of $X \sim S_\alpha(\gamma, \beta, \delta)$,

$$\varphi(t) = \exp\Big( i\delta t - \gamma^{\alpha} \lvert t \rvert^{\alpha} \big[ 1 - i\beta\, \mathrm{sgn}(t) \, w(t) \big] \Big)$$

where $w(t) = \tan\frac{\pi\alpha}{2}$ for $\alpha \ne 1$, and $w(t) = -\frac{2}{\pi}\ln\lvert t \rvert$ for $\alpha = 1$.

The roles of the four parameters:

Parameter	Range	Role
$\alpha$	$(0, 2]$	Tail thickness (smaller means thicker)
$\beta$	$[-1, 1]$	Skewness (positive emphasizes the right tail)
$\gamma$	$(0, \infty)$	Scale
$\delta$	$\mathbb{R}$	Location

The structure becomes clear in the symmetric case ($\beta = 0$, $\delta = 0$):

$$\varphi(t) = \exp\big( -\gamma^{\alpha} \lvert t \rvert^{\alpha} \big)$$

Substituting $\alpha = 2$ gives $e^{-\gamma^2 t^2}$ (Gaussian), and $\alpha = 1$ gives $e^{-\gamma \lvert t \rvert}$ (Cauchy). One expression continuously connects the two distributions.

Three special cases with closed-form densities

Name	$(\alpha, \beta)$	Density
Gaussian	$(2, \cdot)$	$\frac{1}{2\gamma\sqrt{\pi}} e^{-x^2/(4\gamma^2)}$—$\beta$ becomes irrelevant
Cauchy	$(1, 0)$	$\frac{\gamma}{\pi(\gamma^2 + x^2)}$
Lévy	$(\tfrac{1}{2}, 1)$	$\sqrt{\frac{\gamma}{2\pi}}\, \frac{e^{-\gamma/(2x)}}{x^{3/2}}$, $x > 0$—completely skewed to one side

The Lévy distribution describes the time at which Brownian motion first reaches a given level (first hitting time), and is the standard model of "a wait whose mean waiting time is infinite."

Properties: a world governed by $\alpha$

Tails and moments

When $\alpha \lt 2$, the tail decays polynomially (as a power law):

$$P(X > x) \sim C_\alpha (1+\beta) \gamma^{\alpha} x^{-\alpha} \qquad (x \to \infty)$$

Immediate consequences (see the hierarchy of moments):

• $\mathbb{E}[\lvert X \rvert^{p}]$ is finite only for $p \lt \alpha$
• $\alpha \lt 2$: variance is always infinite
• $\alpha \le 1$: even the mean does not exist (the Cauchy belongs here)

The practice of "measuring risk with standard deviation" is fundamentally impossible within this family. $\gamma$ takes over the role of scale, and quantile-based measures such as VaR become practical alternatives.

Additivity (reproductive property)

The sum of independent stable variables with the same $\alpha$ is again a stable variable with the same $\alpha$. In particular, for i.i.d. $X_i \sim S_\alpha$,

$$X_1 + \cdots + X_n \overset{d}{=} n^{1/\alpha} X + d_n$$

It grows as $n^{1/\alpha}$ rather than $\sqrt{n}$. For the Cauchy with $\alpha = 1$, the sum is proportional to $n$—meaning that the sample mean has the same distribution as a single observation. This is precisely the phenomenon in which the LLN becomes powerless.

Self-similarity and Lévy processes

A stochastic process with stable increments (a Lévy flight or $\alpha$-stable Lévy process) is self-similar: $X(ct) \overset{d}{=} c^{1/\alpha} X(t)$. Unlike the continuous paths of Brownian motion ($\alpha=2$), jumps are fundamental when $\alpha \lt 2$—paths consist of sudden leaps, which is why these processes are used to model market crashes, animal foraging (Lévy flight foraging), and anomalous diffusion.

Where are they used?

Area	Phenomenon	Basis
Finance	Heavy tails of log returns (Mandelbrot's study of cotton prices)	Extreme movements occur dozens of times more often than Gaussian predictions
Signal processing	Model of impulsive noise (underwater acoustics and atmospheric noise)	Power-law spikes
Networks	Traffic bursts and file-size distributions	Aggregation of power-law tails
Physics	Anomalous diffusion and Lévy flights	Self-similarity + jumps
ML	Robust learning under heavy-tailed noise and heavy-tail analysis of SGD noise	Empirical studies in which gradient noise exhibits $\alpha \lt 2$

Practical caution: because second-order statistics such as sample variance and sample correlation do not converge in $\alpha \lt 2$ models, estimation uses characteristic-function-based methods (for example, empirical CF fitting), quantile-based methods (McCulloch), or maximum likelihood (numerical density calculation).

Sampling — Chambers–Mallows–Stuck (CMS)

Exact sampling is possible even without a density. In the symmetric case ($\beta = 0$), if $U \sim \mathrm{Uniform}(-\frac{\pi}{2}, \frac{\pi}{2})$ and $E \sim \mathrm{Exp}(1)$ are independent, then

$$X = \frac{\sin(\alpha U)}{(\cos U)^{1/\alpha}} \cdot \Big( \frac{\cos(U - \alpha U)}{E} \Big)^{\frac{1-\alpha}{\alpha}}$$

follows a standard symmetric $\alpha$-stable distribution. Setting $\alpha = 2$ degenerates to the Box–Muller transform—the fact that Gaussian sampling is a special case of CMS is another facet of "Gaussian ⊂ alpha-stable."

Naming caution: SciPy's "alpha distribution" is a different distribution

scipy.stats.alpha is a separate univariate distribution with density $f(x) \propto x^{-2} \exp\big(-\tfrac{1}{2}(a - 1/x)^2\big)$ and is unrelated to the α-stable family on this page. Alpha-stable distributions are provided as scipy.stats.levy_stable. This is an easy source of confusion when moving between the literature and libraries.

Summary of the three distributions — the series conclusion

	Gaussian	Beta	Alpha-stable ($\alpha \lt 2$)
Support	$\mathbb{R}$	$[0,1]$	$\mathbb{R}$ (or a half-line)
Defining axiom	CLT limit / maximum entropy / stability ($\alpha{=}2$)	Conjugacy / order statistics / Gamma ratio	Stability (closure under sums)
Tail	$e^{-x^2/2}$ (ultralight)	None (bounded)	$x^{-\alpha}$ (power law)
Moments	Every order	Every order	Only $p \lt \alpha$
Closed-form density	Yes	Yes	Only three special cases
Fundamental tool	MGF/characteristic function	Beta/Gamma functions	Characteristic function (the only means)
Representative role	Fluctuations of sums, noise	Distribution of probabilities, Bayesian state	Phenomena dominated by extremes

The unifying sentence is: axioms create distributions. Once the generative principle—"what is being summed?" (Gaussian and alpha-stable) or "what is being counted?" (Beta)—is understood, the density formula is only a consequence.

What to remember from this page

1. The single stability axiom $aX_1 + bX_2 \overset{d}{=} cX + d$ determines the entire four-parameter family, and $\alpha \in (0,2]$ governs the tails.
2. There is no closed-form density, but the characteristic function $\exp(-\gamma^\alpha \lvert t \rvert^\alpha [\cdots])$ is always closed form—definition, estimation, and computation all happen in characteristic-function space.
3. Gaussian ($\alpha{=}2$), Cauchy ($\alpha{=}1$), and Lévy ($\alpha{=}\tfrac12$) are the three special cases with closed-form densities.
4. Variance is infinite when $\alpha \lt 2$, and even the mean is absent when $\alpha \le 1$—an entire region where second-order statistical methods collapse.
5. Sums of heavy-tailed data converge to alpha-stable distributions at scale $n^{1/\alpha}$ (generalized CLT)—the Gaussian CLT is its special case.