Regular Variation

Overview¶

An exact Pareto tail has the same probability ratio whenever we multiply a threshold by a fixed amount. Regular variation asks for this behavior only in the limit as the threshold grows. It allows slow corrections to a power law while retaining a limiting tail exponent.

Read the Pareto survival example first for a concrete example. This page uses limits and positive measurable functions; the proof below establishes the power-times-slowly-varying characterization directly from the definition.

Definition and characterization¶

A positive measurable function $f$ is regularly varying at infinity with index $\rho$, written $f\in RV_\rho$, if, for every fixed $t>0$,

The case $\rho=0$ is called slow variation: a positive measurable function $L$ is slowly varying when $L(tx)/L(x)\to1$ for every fixed $t>0$. Equivalently, regularly varying functions can be written as

with $L$ slowly varying. A survival function $\bar F$ has a regularly varying right tail with exponent $\alpha>0$, written $\bar F\in RV_{-\alpha}$, when

Here $\bar F(x)=\mathbb P(X>x)$, $L$ is the slowly varying factor, $\alpha$ is the positive tail exponent, and $t$ is a fixed multiplier.

What the ratio means¶

Regular variation is the mathematical version of Pareto-style scale invariance. At high thresholds, multiplying the threshold by $t$ has an asymptotically stable effect on exceedance probabilities. The slowly varying factor $L$ allows departures from an exact Pareto law while preserving the same tail exponent.

The ratio statement is stronger than saying that large observations are more frequent than under a Gaussian baseline. It says that the relative penalty for raising a large threshold settles to a power $t^{-\alpha}$.

Examples and non-examples¶

• Constant functions are slowly varying. If $L(x)=c>0$, then $L(tx)/L(x)=1$ for every $x$ and $t>0$.

• Logarithmic corrections are slowly varying. For fixed $\beta\in\mathbb R$, $L(x)=(\log x)^\beta$ on $x>1$ satisfies $L(tx)/L(x)\to1$.

• A nonzero power is not slowly varying. If $L(x)=x^\beta$, then $L(tx)/L(x)=t^\beta$, which equals 1 for all $t$ only when $\beta=0$.

• Lognormal right tails are heavy-tailed and subexponential, but not regularly varying: their fixed-multiplier survival ratios do not settle to $t^{-\alpha}$ for any finite $\alpha$.

• Exponential tails are neither regularly varying nor subexponential. For a multiplier $t>1$, their survival ratios decay exponentially in $x$.

Proof of the characterization¶

We prove the algebraic characterization used throughout these notes: a regularly varying function is a power times a slowly varying function. The proof is only an unwinding of the ratio definition. Deeper representation theorems for slowly varying functions are cited through Karamata’s theorem.

If $f(x)=x^\rho L(x)$ with $L$ slowly varying, then for fixed $t>0$,

Conversely, if $f\in RV_\rho$, define $L(x)=x^{-\rho}f(x)$. Then

so $L$ is slowly varying and $f(x)=x^\rho L(x)$.

For survival tails, set $\rho=-\alpha$ in the same equivalence: $\bar F(x)=x^{-\alpha}L(x)$, with $L(x)=x^\alpha\bar F(x)$ slowly varying. No separate survival-tail argument is needed.

For the Pareto distribution with lower cutoff $x_m$,

so $L(x)=x_m^\alpha$ is constant and therefore slowly varying.

Ratio diagnostics¶

The fixed-multiplier ratio can be compared algebraically for an exact Pareto tail and a power law with a slowly varying logarithmic correction.

For an exact Pareto tail and a logarithmically corrected tail, respectively,

For $\alpha=1.5$ and $\beta=1$, the correction multiplier is 2 at $x=2$, $3/2$ at $x=4$, and $5/4$ at $x=16$. The second expression is used only sufficiently far into the tail, where it is decreasing and at most one before taking the ratio; a full distribution also needs a body below that range.

The exact Pareto ratio is constant at $2^{-\alpha}$; the log-corrected ratio approaches the same target slowly. A finite diagnostic curve can suggest regular variation, but the definition is about the asymptotic limit.

Caveats¶

• Regular variation is an asymptotic property. It does not say that every moderate observation follows a power law.

• Estimating $\alpha$ from finite samples is threshold-sensitive; the Hill estimator and log-log plots are diagnostics, not certificates.

• A regularly varying right tail with $\alpha>0$ is subexponential under standard conditions; see Subexponentiality (planned) for the one-big-jump principle.

• Moment implications require assumptions on the full tail and should point to Karamata’s theorem or to a Pareto-specific proof.

References¶

• Bingham, Goldie, and Teugels, Regular Variation 1987.

• Embrechts, Klueppelberg, and Mikosch, Modelling Extremal Events 1997.

• Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

• Used by: Pareto Distribution, Double Pareto Distribution (planned), Subexponentiality (planned), Tail Class Catalog, Pareto Moment Existence, Generalized Central Limit Theorem, and Hill Estimator.

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/theorems/regular-variation.md, revision 9717c9c (2026-09-13 import). Copyright (c) 2023 xshi19. Licensed under MIT. Links and notation were adapted for this site; executable figures and simulations were replaced with static calculations. No upstream execution or formal-proof verification is claimed for this adaptation.