Karamata's Theorem

Statement¶

Let $L$ be eventually positive, measurable, locally bounded, and slowly varying at infinity. If $\rho>-1$, then

If $\rho<-1$, then

Here $a>0$ is fixed in a region where $L$ is locally bounded, and $A(x)\sim B(x)$ means $A(x)/B(x)\to1$. Both integral statements are cited from Bingham et al. (1987), Theorem 1.5.11, p. 28 (direct half); the special case for $L$ and powers also appears as Propositions 1.5.8 and 1.5.10, pp. 26–27.

For a nonnegative random variable $X$ with regularly varying survival function $\bar F(x)=x^{-\alpha}L(x)$ and $\alpha>0$, integration by parts gives the moment-test consequences

and

Thus a Pareto-type tail has finite $p$th moment for $0<p<\alpha$ and infinite $p$th moment for $p>\alpha$. The boundary case $p=\alpha$ is not decided by this theorem alone; it depends on the slowly varying factor $L$.

Here $\bar F(x)=\mathbb P(X>x)$, $\alpha$ is the positive tail exponent, and $p$ is the moment order. The integration parameters $a$ and $\rho$ belong to the integral theorem.

Intuition¶

Karamata’s theorem says that, for regularly varying functions, integrals are asymptotically governed by the endpoint where the mass accumulates. When $\rho>-1$, the integral up to $x$ is controlled by the upper endpoint. When $\rho<-1$, the remaining tail integral beyond $x$ is controlled by the lower endpoint.

This is the bridge between tail shape and moment existence. Once the survival tail behaves like $x^{-\alpha}L(x)$, multiplying by $x^p$ tests whether the $p$th moment is still dominated by ordinary observations or by the far tail.

Examples¶

• For an exact Pareto tail, $L$ is constant, so Karamata reduces the moment test to integrating a pure power.

• For $L(x)=\log x$, the same power boundary applies, but the endpoint asymptotic gains the slowly varying logarithmic factor.

• The boundary case $p=\alpha$ needs separate analysis. Exact Pareto tails diverge logarithmically, as shown in Pareto Moment Existence. More generally,

  $$\mathbb E[X^\alpha]<\infty \Longleftrightarrow \int^\infty \frac{L(t)}{t}\,dt<\infty,$$

  (5)

  when $\bar F(t)=t^{-\alpha}L(t)$ in the tail. Thus $L(t)=1$ gives logarithmic divergence, $L(t)=1/\log t$ gives $\log\log t$ divergence, and $L(t)=1/(\log t)^2$ gives a finite boundary moment.

Deriving the moment consequences¶

We use Karamata’s integral asymptotics as a cited theorem, then derive the two moment consequences by integration by parts. The boundary case $p=\alpha$ is not decided by Karamata alone; it depends on the slowly varying factor and is handled through examples and caveats.

The integral asymptotics are the classical form of Karamata’s theorem for regularly varying functions. The full proof uses uniform convergence of slowly varying functions on compact multiplier intervals and a split of the integral into a near-endpoint part and a negligible remainder; this page cites the standard theorem rather than reproducing that argument here.

The moment consequences follow from integration by parts. For $0<p<\alpha$,

Since $t^{p-1}\bar F(t)=t^{p-\alpha-1}L(t)$ and $p-\alpha-1<-1$, Karamata’s tail-integral form gives

Therefore

For $p>\alpha$,

The part of the integral over any fixed bounded interval is negligible relative to $x^p\bar F(x)$. Applying Karamata to $t^{p-\alpha-1}L(t)$ gives

so subtracting $x^p\bar F(x)$ leaves

For the exact Pareto special case, $L$ is constant and the same formulas are obtained by direct power integration.

Moment-test calculation¶

For a slowly varying example take $L(t)=\log t$ on $t>1$ and $\rho>-1$. Integration by parts gives the exact identity

Dividing by Karamata’s leading term yields

For exact Pareto with $x_m=1$ and $p>\alpha$, the ratio of the truncated moment to its leading term is $1-x^{-(p-\alpha)}\to1$. At $p=\alpha$, the truncated moment is $\alpha\log x$ instead. These are exact calculations illustrating the cited theorem.

Conditional size and the boundary moment¶

For $\alpha>1$, write $e(u)=\mathbb E[X-u\mid X>u]$ for the mean excess. Set $p=1$ in the proved tail-moment consequence and divide by $\bar F(u)>0$. It gives

This is a consequence under a regular-variation assumption, not a finite threshold criterion for fitting that class. It gives the precise forward implication relevant to SCoFT v4 §5.0.2, pp. 93–94 Taleb, 2025. The Pareto note owns the exact version. The chapter guide gives a subexponential Weibull example whose conditional-mean ratio tends to one.

For a complete example with a finite boundary moment, take the survival equal to one below $e$ and, for $x\ge e$, set $\bar F(x)=e^\alpha x^{-\alpha}/(\log x)^2$. It is continuous at $e$ and strictly decreasing afterwards. Tonelli gives

This proves why the exact-Pareto boundary cannot be transferred to every regularly varying tail. For $\alpha=2$ the answer is $3e^2$, whereas an exact Pareto of exponent 2 has infinite second moment.

The chapter 5 integral figure compares the logarithmic integral in the earlier moment-test calculation with its Karamata leading term. The demo also evaluates the boundary example in logarithmic coordinates, integrating to infinity.

Caveats¶

• The assumptions on $L$ matter. Slow variation is an asymptotic regularity condition, not a finite-sample diagnostic.

• Karamata does not settle the boundary moment $p=\alpha$. Exact Pareto tails diverge logarithmically there, while other slowly varying factors can change the boundary behavior.

• Moment existence is a tail statement. Estimating a moment from data also depends on sample size, dependence, threshold choice, and whether the fitted tail model is credible.

• We treat nonnegative right tails. Two-sided models need the same reasoning applied to $|X|$ or to each side separately.

References¶

• Bingham et al. (1987), Regular Variation, Theorem 1.5.11, p. 28; Propositions 1.5.8 and 1.5.10, pp. 26–27. These support the cited integral asymptotics; the moment and conditional-mean consequences are derived here.

• Taleb (2025), §§5.0.2 and 5.1.1, pp. 93–94 and 95–97: conditional-size and scaling context, not the proof source for Karamata.

• Feller, An Introduction to Probability Theory and Its Applications, Vol. II (1971, 2nd ed., Wiley).

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

• Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

• Depends on: Regular Variation and the canonical tail notation in Shared notation.

• Used by: Pareto Distribution, Pareto Moment Existence, LLN Failure Under Infinite Mean, Max-to-Sum Ratio, and Hill Estimator.