Lévy Stable Distributions

The normal distribution has a special property that makes it the default model for sums of noise: add two independent normals together, and the result is still normal (just with combined mean and variance). Very few distributions have this "stability" property. The Lévy stable family is the entire list of distributions that do — and the normal turns out to be just one special, tame member of a much wilder family, most of which have infinite variance and were proposed decades ago as a better fit for real financial returns than the normal ever was.

The analogy: a family recipe that always reproduces itself

Some recipes, no matter how many times you combine batches, always yield the same style of dish, just a bigger portion — that's "stability." Stable distributions are the probability equivalent: sum any number of independent, identically distributed draws from a stable distribution, and (after simple rescaling) you get back a draw from the same family, just with different scale and location. The normal distribution does this. So does the Cauchy distribution. So does every member of a broader four-parameter family — and Mandelbrot's famous 1963 observation was that real cotton price changes fit the wilder, fat-tailed members of this family far better than they fit the normal.

Writing it down

A distribution is stable if, for independent copies $X_1, X_2$ of $X$ and any $a,b>0$, there exist $c>0, d$ such that

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

In words: a weighted sum of two independent draws from the distribution has the same shape as a single (rescaled, shifted) draw from that same distribution. Stable distributions don't generally have a closed-form density, but are described by four parameters via their characteristic function: $\alpha \in (0,2]$ (the tail index — smaller means fatter tails), $\beta \in [-1,1]$ (skewness), $\gamma > 0$ (scale), $\delta$ (location). $\alpha = 2$ gives exactly the normal distribution (finite variance, thin tails); $\alpha = 1, \beta=0$ gives exactly the Cauchy distribution (no mean, no variance); any $\alpha < 2$ means infinite variance, and any $\alpha \le 1$ means infinite mean too (unless $\beta$ pushes support one-sided). The generalized CLT says: sums of i.i.d. variables with infinite variance, appropriately rescaled, converge not to a normal but to a stable distribution with matching $\alpha$ — the ordinary CLT is the special case where the underlying variance is finite, forcing $\alpha=2$.

This is the $\alpha=2$ member of the family — the only one with a familiar bell shape and finite variance. Every other value of $\alpha$ produces a curve that looks similarly bell-shaped near the center but has visibly heavier tails the smaller $\alpha$ gets, exactly like moving from a normal toward the Cauchy shown in that page.

Worked example 1: reading the tail index off return data

Suppose 5-minute equity returns are fit to a stable distribution and the estimated tail index is $\hat\alpha = 1.7$. Since $\hat\alpha < 2$, the theoretical variance of this fitted model is infinite — meaning the sample variance you'd compute from a finite dataset keeps growing, never settling down, as you add more data (a phenomenon real high-frequency return data does sometimes exhibit). Compare $\hat\alpha=1.9$ (closer to normal, thinner tails, but still technically infinite variance) versus $\hat\alpha=1.2$ (much fatter, closer to Cauchy-like, arguably infinite mean too) — the qualitative behavior of extreme moves in a simulation from each is dramatically different even though both are "stable" and both would look roughly bell-shaped on a casual histogram.

Worked example 2: aggregating over time doesn't normalize fat tails

Under the ordinary CLT, aggregating $n$ independent daily returns into an $n$-day return makes the sum look increasingly normal. Under a stable model with $\alpha=1.7$, aggregating $n$ returns instead produces another stable distribution with the same $\alpha=1.7$ — rescaled by $n^{1/\alpha}$ instead of the normal CLT's $n^{1/2}$. Concretely, 100-day aggregated volatility scales like $100^{1/1.7} \approx 18.7\times$ the 1-day scale, versus $100^{1/2}=10\times$ under a normal assumption — the stable model predicts nearly twice as much accumulated dispersion over 100 days as the normal model does, and critically, the shape stays fat-tailed no matter how long you aggregate. This directly contradicts the common (and often false) assumption that longer-horizon returns are "more normal."

A stable distribution's tail also decays as a power of $x$ (like this curve), with exponent set by $\alpha$ — this is the same power-law tail behavior as the Pareto and Cauchy distributions, which is exactly why $\alpha<2$ stable distributions inherit their infinite-variance problem.

What this means in practice

• Risk model choice. If tail index estimates from your return data sit noticeably below 2, a normal-based VaR model is understating tail risk by construction, and time-aggregation (annualizing by $\sqrt{n}$) is using the wrong scaling exponent entirely.
• Mandelbrot's original argument. The historical motivation for this family was exactly this: observed cotton and stock price changes had far more extreme moves than a normal model predicted, at every timescale — not something that "averages out."
• Estimation is hard. Stable distributions (outside $\alpha=1,2$) have no closed-form density, so fitting requires specialized methods (characteristic function matching, quantile methods) rather than ordinary maximum likelihood.