Pareto Moment Existence

Statement¶

Let $X$ have a Pareto Type I distribution with lower cutoff $x_m>0$ and tail exponent $\alpha>0$ :

\bar F(x)=\mathbb P(X>x)=\left(\frac{x_m}{x}\right)^\alpha, \qquad x\ge x_m.

(1)

For any moment order $p>0$ ,

\mathbb E[X^p] < \infty \quad\Longleftrightarrow\quad p<\alpha.

(2)

When $p<\alpha$ ,

\mathbb E[X^p]=\frac{\alpha x_m^p}{\alpha-p}.

(3)

At $p=\alpha$ , the truncated moment diverges logarithmically. For $p>\alpha$ , it diverges as a power of the upper cutoff. In particular, the Pareto mean exists only for $\alpha>1$ , and the variance exists only for $\alpha>2$ .

Here $\bar F$ denotes survival probability and $\mathbb E$ denotes expectation; $p$ is a positive moment order.

Proof by direct integration¶

We prove the Pareto Type I moment boundary and the moment formula by direct integration. The same calculation also explains the logarithmic boundary and power-divergent regimes in the truncated-moment section. The broader regularly varying moment test is handled by Karamata’s Theorem.

For $x\ge x_m$ , the Pareto density is

f(x)=\alpha x_m^\alpha x^{-(\alpha+1)}.

(4)

For $p>0$ ,

\mathbb E[X^p] = \int_{x_m}^{\infty}x^p\alpha x_m^\alpha x^{-(\alpha+1)}\,dx = \alpha x_m^\alpha \int_{x_m}^{\infty}x^{p-\alpha-1}\,dx.

(5)

The integral converges exactly when $p-\alpha-1<-1$ , equivalently $p<\alpha$ . Evaluating the convergent case gives

\alpha x_m^\alpha \cdot\frac{x_m^{p-\alpha}}{\alpha-p} = \frac{\alpha x_m^p}{\alpha-p}.

(6)

Adding $\alpha+1$ to both sides proves the algebraic equivalence $p-\alpha-1<-1 \Longleftrightarrow p<\alpha$ . Lean formalization is planned; the integral criterion and evaluation here are an ordinary proof.

This exact Pareto calculation is the simplest instance of the Karamata moment test for regularly varying tails.

Truncated-moment behavior¶

For a finite upper cutoff $b\ge x_m$ , define

M_p(b)=\mathbb E[X^p\mathbf 1_{\{X\le b\}}].

(7)

Direct integration gives

M_p(b)= \begin{cases} \dfrac{\alpha x_m^\alpha}{\alpha-p} \left(x_m^{p-\alpha}-b^{p-\alpha}\right), & p<\alpha,\\[1.1em] \alpha x_m^\alpha\log(b/x_m), & p=\alpha,\\[0.8em] \dfrac{\alpha x_m^\alpha}{p-\alpha} \left(b^{p-\alpha}-x_m^{p-\alpha}\right), & p>\alpha. \end{cases}

(8)

Thus the same theorem has three operational regimes: convergence to a finite moment, logarithmic boundary growth, and power growth.

For $\alpha=1.5$ and $x_m=1$ , the three regimes reduce to:

Moment order $p$	Truncated moment $M_p(b)$ , $b\ge1$	Limit as $b\to\infty$
1	$3(1-b^{-1/2})$	3
1.5	$1.5\log b$	Infinite, logarithmic growth
2	$3(\sqrt b-1)$	Infinite, power growth

What to notice. For $p<\alpha$ , the truncated moment levels off. At $p=\alpha$ , it keeps growing, but only logarithmically. For $p>\alpha$ , the upper tail contributes a visible power-law rise.

Examples¶

If $\alpha=0.8$ , the mean is infinite because $1\ge\alpha$ .
If $\alpha=1.5$ , the mean exists, but the variance is infinite because $2\ge\alpha$ .
If $\alpha=3$ , the mean and variance exist, while the third raw moment is at the logarithmic boundary and diverges.

The theorem is exact, but sample estimates can still look misleading. With $\alpha$ close to a boundary, a finite run can appear calm until a new large observation changes the empirical moment.

For $\alpha=1.2$ and $x_m=1$ , the population mean is 6 while the second raw moment is infinite. Any finite sample of finite observations still has a finite sample second moment. Its finiteness does not settle the population question.

Caveats¶

Moment existence is a property of the generating distribution, not proof that a finite sample estimate will be accurate.
When $\alpha$ is close to a boundary, convergence can be so slow that the formal moment is a poor operational summary.
For two-sided heavy-tailed variables, check absolute moments or one-sided tails explicitly. Symmetry can make a location parameter look finite while absolute exposure is infinite.
Empirical Pareto fits require threshold checks. A moment calculation using a fitted $\widehat\alpha$ inherits the uncertainty and bias of the tail fit.

References¶

Bingham, Goldie, and Teugels, Regular Variation 1987.
Resnick, Heavy-Tail Phenomena 2007.

Backlinks¶

Depends on: Pareto Distribution and Karamata’s Theorem.
Used by: LLN Failure Under Infinite Mean, Pre-Asymptotic LLN Behavior (planned), and Max-to-Sum Ratio (planned).

Source and adaptation¶

Adapted from incerto-wiki, content/concepts/theorems/pareto-moment-existence.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.

References¶

Bingham, N. H., Goldie, C. M., & Teugels, J. L. (1987). Regular Variation. Cambridge University Press. 10.1017/cbo9780511721434
Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. In Springer Series in Operations Research and Financial Engineering. Springer New York. 10.1007/978-0-387-45024-7