Symmetric stable distributions

You will learn: Check sum scaling and moment existence for symmetric stable distributions.

Start with: Cauchy distribution · Empirical CDF

Does averaging always reduce spread at the usual square-root rate? The normal distribution says yes under its model; the Cauchy distribution gives a striking counterexample. Both belong to the stable family. This lesson explores its symmetric, zero-location members. It does not cover skewed stable laws or all their parameter conventions.

A distribution is stable when sums of independent copies have its shape after a suitable rescaling and, in general, recentering. In this symmetric zero-location family, independent copies satisfy an exact identity in distribution:

\frac{X_1+\cdots+X_n}{n^{1/\alpha}}\overset{d}{=}X_1,\qquad 0<\alpha\le2.

The symbol means equality of distributions, not equality of paired simulated values. The demo supports α from 0.5 to 2 and compares complete simulated blocks using empirical CDFs.

Compare one observation with

Solid blue: first observation from each block. Dashed red: normalized block sum. Both are empirical CDFs from 400 blocks. Their denominators remain 400, even outside the window. The two curves share observations and are not independent simulations.

Distribution	Population scale	Empirical median	Empirical interquartile width	Below / above window
One observation	1.00000	-0.0928671	1.90983	7 / 9
Sum / n^(1/α)	1.00000	-0.00624298	2.04354	7 / 6
Ordinary average	0.396850	-0.00247753	0.810980	1 / 3

The ordinary average has scale c·n^(1/α−1) = 0.3968503. It shrinks with n. This describes distributions, not monotone behavior along a particular sample path.

Stability index α 1.50 Population scale c 1.00 Observations per block n 16.00 Simulated blocks 400.00 Window half-width in units of c 5.00 Random seed 43.00

Population moments do not follow from a finite sample

One observation	Population quantity
Mean	0
Second moment E[X²]	Infinite
Special case	Symmetric, zero-location α-stable

All generated values enter the sums and quantiles; windowing only changes the display. This implementation uses finite-precision pseudorandom numbers and cannot sample arbitrarily remote mathematical tails. A finite sample mean or second moment does not establish that the corresponding population moment exists.

Check the characteristic function and inspect the first block

For this symmetric family, E[cos(tX)]=exp(−|ct|^α). Unlike X itself, cos(tX) is bounded. The sample average below is a simulation check, not a fitted parameter or a proof of stability.

Characteristic frequency t 0.50

Distribution	Exact E[cos(tX)]	Simulated average
One observation	0.7021885	0.6838946
Sum / n^(1/α)	0.7021885	0.6955477
Ordinary average	0.9154053	0.9158914

First-block observation	Value
1	0.7360974
2	-0.4308312
3	0.2106310
4	-0.2919120
5	-2.010921
6	-1.742657
7	-2.190243
8	-0.8790042
9	-1.376402
10	5.963415
11	0.4769076
12	0.1909820
13	1.875765
14	0.3677379
15	-0.4755098
16	-0.7384572

The experiment covers symmetric stable laws (skew parameter zero) with zero location and characteristic function exp(−|ct|^α). Increasing n preserves each block's existing observations; increasing the block count preserves existing blocks.

Fix the scale convention first

This lesson defines scale c by the characteristic function:

\varphi_X(t)=E[e^{itX}]=\exp(-|ct|^\alpha).

Symmetry makes its imaginary part zero, so E[cos(tX)] is the real expression shown above. This bounded expectation exists even when E[X] does not. It also supplies a numerical simulation check that does not rely on estimating nonexistent moments.

At α=2, the model is normal with mean zero and variance 2c². Thus c is not its standard deviation. At α=1, it is Cauchy with scale c. For other displayed α it is another symmetric stable law; “stable” describes its behavior under addition, not the visual steadiness of a sample average.

Derive the scaling from independence

Characteristic functions multiply for independent sums. If Sₙ=X₁+…+Xₙ, then:

\varphi_{S_n}(t)=\bigl[\exp(-|ct|^\alpha)\bigr]^n=\exp\bigl(-|c n^{1/\alpha}t|^\alpha\bigr).

So Sₙ has scale c n¹ᐟᵅ. Dividing by n¹ᐟᵅ returns scale c and gives the exact stability identity. Dividing by n instead produces the ordinary average, with scale:

c_{\bar X_n}=c n^{1/\alpha-1}.

For n=16 and c=1, α=2 gives scale 1/4; α=1 gives 1; and α=0.5 gives 16. At the default α=1.5, the average scale is 16⁻¹ᐟ³≈0.396850. Increasing n narrows the average’s distribution for α>1, leaves it unchanged for α=1, and widens it for α<1. None of those statements requires a particular simulated path to move monotonically.

Make a prediction

With α=0.5, should dividing a 16-observation sum by 16 recover the original distribution?

Explore the answer

No. The stability normalization is 16^(1/0.5)=256. Dividing only by 16 leaves a scale 16 times the original. Switch between normalized sum and ordinary average to compare both with the same underlying simulated blocks.

Moments require more than symmetry

For 0<α<2, the absolute moment E[|X|ᵖ] is finite for 0<p<α and infinite for p≥α. Consequently, the symmetric mean exists and is zero for α>1; for α≤1 it is undefined. Positive and negative divergent integrals do not cancel into a valid expectation. The second moment is infinite for every α<2. The normal case at α=2 has finite moments and is an exception to this power-tail rule.

The empirical medians and interquartile widths in the table remain useful finite-sample descriptions. They are not exact population quantiles. A finite displayed sample mean or variance cannot establish that the corresponding population quantity exists.

The usual finite-variance Central Limit Theorem is not contradicted. Its assumptions do not hold for these non-Gaussian stable examples. Generalized limit theorems need their own tail and normalization conditions; an arbitrary heavy-tailed distribution is not automatically an exactly stable one.

What the simulation does

Independent uniform and exponential variates feed the symmetric Chambers–Mallows–Stuck construction. Each block has a repeatable random stream, so increasing n retains its previous observations. Increasing the block count retains all earlier blocks. The blue curve uses the first observation of each block, so it shares data with the red curve; their Monte Carlo discrepancies are not independent.

The display window can omit large values visually, but every value remains in sums, quantiles, and ECDF denominators. Counts outside the window are shown. Pseudorandom finite-precision arithmetic cannot reach arbitrarily remote mathematical tails. This experiment therefore illustrates and checks the specified model rather than proving it from its simulated pictures.

Make a prediction

If two empirical CDFs overlap closely, have you proved stability for an observed real-world process?

Explore the answer

No. Here the sampler was constructed for a stable model whose identity is derived separately. For observed data, approximate agreement could reflect sampling noise, a limited window, or another distribution. Dependence and the sampling mechanism also need investigation.

Sources

SciPy’s stable-distribution reference specifies the characteristic-function convention and normal/Cauchy special cases. Random Services: stable distributions develops stability under sums. John Nolan’s introductory chapter, sections 1.5 and 1.7, gives the moment conditions and simulation construction. Numerical checks compare the sampler with pinned SciPy CDFs and bounded characteristic-function expectations.

Return to learning paths and foundations, or compare Zipf’s finite versus infinite support.

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