Heavytailed

Heavy-tailed describes probability distributions whose tails are heavier than those of the exponential distribution. In practical terms, extreme values occur more frequently under a heavy-tailed model than under light-tailed ones like the normal distribution. A common formalization uses tails that decay polynomially: P(X>x) ~ L(x) x^{-α} as x→∞, where α>0 and L is a slowly varying function. The tail index α governs heaviness: smaller α indicates fatter tails. When α≤1, the mean is infinite; when α≤2, the variance is infinite.

Several well-known distributions are heavy-tailed. The Pareto distribution has a power-law tail; the Cauchy distribution is undefined in its mean and variance; t-distributions with low degrees of freedom also exhibit heavy tails. The lognormal distribution is also considered heavy-tailed, as are certain Weibull models with shape parameter less than one. These tails imply higher risk of very large observations than light-tailed models.

Applications and implications of heavy tails are widespread. They appear in finance for large market moves, in insurance for big claims, in internet traffic and file-size distributions, and in environmental and natural phenomena where extremes matter. Statistical analysis under heavy tails often relies on extreme value theory, with methods to estimate tail heaviness such as tail index estimators, and risk measures geared to extreme outcomes. Standard techniques assuming finite variance or normality can perform poorly, motivating robust or tail-aware approaches.

Terminology varies: heavy-tailed is sometimes equated with fat-tailed, though some authors distinguish nuanced meanings among fat-tailed, subexponential, and regularly varying categories.