A single exponent alpha parametrizes a family spanning normal to wildly fatal-tailed distributions

Evidence

The parameter that is most important in distinguishing between different members of the Levy family of distributions is the “characteristic exponent,” which Levy denoted by the Greek letter a (alpha). Alpha is always greater than zero and no greater than 2. The lower the value of alpha, the more cases fall in the tails of the distribution: in other words, the higher the proportion of events that are extreme, “wild,” deviations from the average. — Donald MacKenzie, An Engine, Not a Camera, 479

Categorization

• DEFINE: Introduces the characteristic exponent alpha, the parameter that sorts Levy distributions from normal to fat-tailed.
• XREF: The alpha parameter and fat tails directly connect to Taleb’s Black Swan work and Mandelbrot’s own ‘mild vs wild randomness’ vocabulary. — connects to: Under a Gaussian distribution extreme tail events become effectively impossible — Mark Blyth, Austerity: The History of a Dangerous…, 106; A finite sample’s variance cannot distinguish a fat-tailed from an infinite-variance distribution — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 499; The Levy distribution’s failure to survive time aggregation cooled academic interest in fat tails — Donald MacKenzie, An Engine, Not a Camera: How Financial…, 509 (+3 more)

Broader Connections

• A finite sample’s variance cannot distinguish a fat-tailed from an infinite-variance distribution #expands (0.85) — Donald MacKenzie, An Engine, Not a Camera, 499
• Levy distributions yield more stable estimates of extreme moves than variance-based methods #exemplifies (0.82) — Donald MacKenzie, An Engine, Not a Camera, 937
• The Levy distribution’s failure to survive time aggregation cooled academic interest in fat tails #complicates (0.80) — Donald MacKenzie, An Engine, Not a Camera, 509
• Stock price changes are modeled with thin-tailed lognormal distributions, not fat-tailed ones #contradicts (0.80) — Donald MacKenzie, An Engine, Not a Camera, 138
• Empirical aggregation tests undercut the Levy distribution’s self-similarity claim #complicates (0.75) — Donald MacKenzie, An Engine, Not a Camera, 507