Cauchy Sample-Mean Stability

Theory

A Cauchy variable with location parameter μ and scale parameter γ has the probability density function (PDF):

f(x) = γ / (π * ((x - μ)² + γ²))

The characteristic function is φ(t) = exp(i * μ * t - γ * |t|). When we average n independent and identically distributed Cauchy variables, the characteristic function of the sample mean is:

φ_mean(t) = (φ(t / n))ⁿ = (exp(i * μ * t / n - γ * |t / n|))ⁿ = exp(i * μ * t - γ * |t|)

This is precisely the characteristic function of the original Cauchy(μ, γ) distribution. Thus, the sample mean is strictly stable. Its scale remains γ regardless of n. This contrasts sharply with a normal distribution, where the standard error scales as 1/√n. Since the Cauchy expectation is mathematically undefined (the integral for the mean does not absolutely converge), the location parameter μ behaves as a median rather than a finite population mean.

Experiment

Use the interactive module to compare the stability of the Cauchy mean against a hypothetical normal sample mean that matches the scale parameter.
Link to the interactive module: https://riano.app/module/cauchy-sample-mean-stability

1. Change the Sample size from 1 to 4, then to 16. Notice that the Cauchy Mean PDF curve remains unchanged. The Cauchy scale stays exactly the same.
2. Observe the Normal comparison density curve. Its peak gets narrower and taller as you increase the sample size because its standard error shrinks by 1/√n.
3. Adjust the Window radius to see how the probability mass around the center stays constant for the Cauchy distribution.

Worked Example

Let's calculate the centered window probability for a Cauchy distribution with scale parameter γ = 1 and window radius a = 1.

1. The formula for the probability in a symmetric window around μ is: P(μ - a < X < μ + a) = 2 * arctan(a / γ) / π
2. Substitute a = 1 and γ = 1: P = 2 * arctan(1) / π
3. We know arctan(1) = π / 4
4. Therefore, P = 2 * (π / 4) / π = 2 / 4 = 0.5

Expected result: 0.5

Limitations

The Cauchy distribution is a mathematical idealization. In real-world physical measurements, devices always have bounded ranges, effectively truncating the heavy tails. While true infinite variance is rare in reality, the Cauchy distribution serves as a crucial counterexample in probability theory to demonstrate what happens when the assumptions of the Law of Large Numbers and Central Limit Theorem are violated. Additionally, since the true expectation is undefined, interpreting the location parameter as an arithmetic mean is fundamentally incorrect; robust statistics like the sample median must be used instead.

References

More information about the stable distribution class and the properties of the Cauchy distribution can be found below:

• https://www.randomservices.org/random/special/Stable.html
• https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.cauchy.html