The Cauchy Distribution and Heavy Tails

The Cauchy distribution (f(x) = 1/(π(1+x²))) has no mean or variance; its tails decay so slowly that sample averages never converge (they follow the same distribution). This violates the law of large numbers. It appears in physics as the Lorentzian line shape (spectral lines) and in finance as an example of extreme risk. Unlike the normal distribution, the Cauchy can produce outliers arbitrarily far from the centre. Its median is well‑defined, but its mean is undefined. It serves as a warning against blindly assuming normality.