An asymptotic property of positive functions at infinity (or at zero) characterized by power-law scaling: a measurable function f is regularly varying at infinity with index α if for every x>0, lim_{t→∞} f(tx)/f(t) = x^α. When α=0 the function is slowly varying.

Principle

Regular variation encodes asymptotic self-similarity under multiplicative rescaling, reducing tail or large-argument behavior to a combination of a power law t^α and a slowly varying factor; it is the organizing idea behind Karamata theory and Tauberian relations.

Demonstration

A common example is f(t)=t^α L(t) for α∈R with L slowly varying, e.g. L(t)=log(t)^{β}. In probability, heavy-tailed distributions satisfy P(X>t) ∼ t^{-α}L(t), giving power-law tails used in limit theorems.

Misapplication

Assuming regular variation from finite-scale observations or fitting a power law over a narrow range; treating oscillatory factors that do not settle asymptotically as regular variation; or using it to claim exact equality rather than asymptotic equivalence.

Consequence

Regular variation yields uniform asymptotic relations (Karamata’s representation, Potter bounds), enables precise tail approximations, governs domains of attraction for stable laws, and allows Tauberian theorems to relate transforms and tails.

Reversal

Rapid variation, where f(tx)/f(t) → ∞ or 0 (no finite power-law index), or simple bounded variation that lacks multiplicative scaling; or functions with essential oscillation that prevent a limit ratio from existing.

Boundary

Defined for positive measurable functions near infinity or zero; it excludes sign-changing functions unless one applies modulus or separates positive and negative parts, and it does not assert global power-law behavior away from the asymptotic regime.

Semantic Tension

Tension exists with colloquial 'power law' fits and with mere asymptotic equivalence: regular variation is a specific limiting property under multiplicative scaling, stronger than observing approximate algebraic decay and distinct from slow decay or oscillatory asymptotics.

Synthesis

Regular variation formalizes asymptotic power-law scaling by factoring a function into a power term and a slowly varying multiplier, providing uniform tools (representation, bounds, Tauberian links) to turn multiplicative self-similarity into precise asymptotic conclusions.