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. The Karamata note develops the related integral asymptotics and their moment consequences.

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.

What the representation theorem adds¶

The elementary equivalence above defines $L$; it does not prove Karamata’s representation theorem. For positive measurable slowly varying $L$, that theorem gives, eventually,

with measurable $c$ and locally integrable $\varepsilon(v)/v$. We cite the existence of this representation: Bingham et al. (1987), Theorem 1.3.1, p. 12. This is a different result from the integral theorem.

Its sufficiency follows directly. For fixed $t>0$, the logarithm of $L(tx)/L(x)$ is $\log[c(tx)/c(x)]+\int_x^{tx}\varepsilon(v)\,dv/v$. The integral’s absolute value is at most $|\log t|$ times the supremum of $|\varepsilon|$ between $x$ and $tx$, which tends to zero. Also $\log L(x)/\log x\to0$: split the integral at a fixed point beyond which $|\varepsilon|<\delta$, divide by $\log x$, and then let $\delta\downarrow0$. Hence a regularly varying survival satisfies $\log\bar F(x)/\log x\to-\alpha$. This is a ratio of logarithms; convergence of a derivative of a log-log plot needs extra smoothness.

Positive powers of a tail variable¶

We prove the transformation rule used in SCoFT v4 §5.2.3, p. 100 Taleb, 2025, without differentiating a density. Let $X\ge0$ have $\bar F(x)=x^{-\alpha}L(x)$ eventually and let $r>0$. Then

For fixed $t>0$, the ratio of the last slowly varying factors has arguments $t^{1/r}y^{1/r}$ and $y^{1/r}$, so it tends to one. Thus $X^r$ has survival exponent $\alpha/r$. Units may be fixed first by applying this argument to $X/x_m$. Negative powers concern the behavior near zero and are outside this right-tail rule. The exact Pareto payoff case is already derived in asymmetric payoffs.

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.

Use regular_variation_ratio to evaluate the law used in the figure:

import numpy as np
from xmath.incerto.tail_diagnostics import regular_variation_ratio

survival = lambda u: u**-2 * (1 + np.log(u))
result = regular_variation_ratio(survival, [1, 10, 100, 10000])
for u, ratio in zip(result["threshold"], result["ratio"]):
    print(f"u={u:.4g}: ratio={ratio:.4g}, limit=0.25")

Output

u=1: ratio=0.4233, limit=0.25
u=10: ratio=0.3025, limit=0.25
u=100: ratio=0.2809, limit=0.25
u=1e+04: ratio=0.267, limit=0.25

The existing producer is uv run python demos/incerto/glossary_worked_cases.py.

The chapter 5 demo compares multiplier ratios for Student and shifted double-Pareto tails. Their finite threshold differences are explained by the matching calculation.

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 nonnegative law with a regularly varying right tail and $\alpha>0$ is subexponential (Foss, Korshunov, and Zachary, 2nd ed., §3.5, Theorem 3.29 and following paragraph, p. 57); see Subexponentiality 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¶

• Foss, Korshunov, and Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, 2013, 2nd ed., §3.5, Theorem 3.29, p. 57: cited subexponential classification.

• Bingham et al. (1987), Regular Variation, Theorem 1.3.1, p. 12: the cited representation theorem; the elementary power-times-$L$ characterization is proved on this page.

• Taleb (2025), §5.1.1, pp. 95–97, and §5.2.3, p. 100: scaling and transformation context.

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

• Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

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