Pareto Moment Existence

Statement¶

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

For any moment order $p>0$,

When $p<\alpha$,

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

For $p>0$,

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

Adding $\alpha+1$ to both sides proves the algebraic equivalence $p-\alpha-1<-1 \Longleftrightarrow p<\alpha$.

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

Direct integration gives

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:

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.

Products and moment integrals¶

Use pareto_truncated_moment.

from xmath.incerto.power_laws import pareto_truncated_moment

for order in (1, 1.5, 2):
    moments = pareto_truncated_moment([10, 100, 10000], alpha=1.5, order=order)
    print(f"order={order:.4g}: [" + ", ".join(f"{v:.4g}" for v in moments) + "]")

Output

order=1: [2.051, 2.7, 2.97]
order=1.5: [3.454, 6.908, 13.82]
order=2: [6.487, 27, 297]

These are indicator-truncated population moments; no sample is censored.

Figure regeneration: uv run python demos/incerto/power_laws.py. Each panel uses 241 cutoffs from 1 to 10⁴ (the middle omits the zero endpoint, leaving 240 points; the last begins at 1.01 to avoid a zero leading term). The Pareto product proof and Karamata calculation own the other two formulas. The cutoff is the indicator truncation in $M_p(b)$, not conditioning on $X\le b$ and not the capped moment $\mathbb E\min(X,b)^p$.

The chapter’s transformation discussion (§5.2.3, p. 100) and squared Student example (§5.6, pp. 103–105) use different moment orders Taleb, 2025. The moment-estimation note already explains why an existing second moment need not give a finite variance to the empirical average of squares.

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¶

• Taleb (2025), §§5.2.3 and 5.6, pp. 100 and 103–105: power transformations and squared observations; the exact Pareto integral and its equality boundary are proved here.

• Bingham et al. (1987), Regular Variation.

• 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, and Max-to-Sum Ratio.