regularly varying tails

The fine line between the Gaussian world and the stable/heavy-tailed world is drawn by the rate at which a distribution's tail decays. Regular variation is the precise asymptotic language for power-law tails. It is the exact condition that places a law in the domain of attraction of a non-Gaussian stable law, and it is the backbone of extreme value theory.

A positive function L on (0, infinity) is slowly varying if L(cx)/L(x) -> 1 as x -> infinity for every fixed c > 0 (examples: constants, log x, log log x — anything that grows or decays slower than any power). A function f is regularly varying with index rho if f(x) = x^rho L(x) for a slowly varying L; equivalently f(cx)/f(x) -> c^rho. A distribution has regularly varying (right) tails of index -alpha (alpha > 0) if its tail P(X > x) = x^(-alpha) L(x) for slowly varying L. The single number alpha is the tail index; it controls which moments exist: E[X^p] < infinity for p < alpha and = infinity for p > alpha (the boundary p = alpha depends on L). The Karamata theorems make this rigorous, letting you integrate and differentiate regularly varying functions asymptotically and connecting tail integrals to the tail itself.

Regularly varying tails matter because they are the EXACT membership criterion for the domain of attraction of an alpha-stable law (alpha < 2): summing such variables and normalizing by n^(1/alpha) gives a stable limit. They also govern extreme-value behaviour (the Frechet domain), large claims in insurance, and the difference between thin and fat tails in risk. Honest caveats: regular variation is purely an asymptotic (large-x) statement and says nothing about the body of the distribution, so two laws can match in the bulk yet differ utterly in attraction class. The slowly varying factor L genuinely matters at the boundary (it determines the exact norming sequence and whether the boundary moment is finite), and 'heavy-tailed' is broader than 'regularly varying' — lognormal and Weibull-type tails are heavy but NOT regularly varying, and they sit in the Gaussian (or Gumbel) domain, not the stable one.