Chapter 5 — Subexponentials and power laws

Chapter 5 compares two different tail properties. Subexponentiality describes a large sum of a fixed number of independent observations. Regular variation describes how survival changes when we multiply a threshold. The latter implies the former for nonnegative laws with positive tail exponent, but the converse fails.

We use Taleb (2025), third edition (2025), arXiv:2001.10488v4, chapter 5, printed pp. 91–105, and its attached appendix B, pp. 127–133. The notes develop their own examples and arguments. The source locators below identify context; each concept page identifies its proved and cited results.

Reading map¶

Keep the three ratios separate¶

For iid nonnegative $X_1,X_2$ with survival $\bar F(u)=P(X>u)>0$, define the following local ratios:

Regular variation asks whether $A_t(u)\to t^{-\alpha}$ for every fixed $t>0$. Subexponentiality asks whether $R(u)\to1$. The event comparison $Q_F$ alone supplies neither definition. For an exponential law, $Q_F=1$ identically, while $R(u)=(1+\lambda u)/2$ diverges. The ratio note gives exact calculations and explains why translating a light-tailed variable changes the limit of $Q_F$.

For a Pareto law, the fixed-size rare-event limit holds for every $\alpha>0$. It does not say that the maximum accounts for nearly all of an increasingly long sample. The existing Pareto regimes separate $n\to\infty$ from $u\to\infty$ with $n$ fixed, including the infinite-mean boundary $\alpha=1$.

Exponents need support and dependence assumptions¶

For nonnegative variables and positive fixed weights, a union bound and a one-coordinate lower bound preserve the smallest logarithmic tail exponent of a finite sum. That statement does not require independence. Signed variables can cancel: $(V,-V)$ has sum zero even when both marginals have power tails. The sum proof states exactly what the bounds establish.

Products are more sensitive to dependence. Two independent unit-cutoff Pareto variables with common exponent $\alpha$ have product survival

The product derivation also handles unequal exponents. With identical observations $X_2=X_1$, the product is a square and its exponent is $\alpha/2$ instead. Thus the unqualified product principle in §5.2 cannot be used under arbitrary dependence.

For a nonnegative variable with regularly varying survival and $r>0$, $X^r$ has exponent $\alpha/r$. Applying this to $|T_\nu|$, where $T_\nu$ is a Student variable with degrees of freedom $\nu>0$, gives exponent $\nu/2$ for $T_\nu^2$. Its mean exists for $\nu>2$ and its variance for $\nu>4$. These moment conditions do not identify a stochastic-volatility process from a fluctuating sample statistic. Chapter 4 already supplies the estimator-moment distinction. The special-function expression and mean-deviation table in §5.6 are not needed for this argument and are not claimed verified here.

Qualifications to carry into the source¶

• Exponential and gamma laws fail the convolution-subexponential criterion; the catalog’s direct proof takes precedence over treating them as members of that class.

• A slowly varying factor need not approach a positive constant. The existing counterexample and the equal-exponent product above retain a logarithmic factor forever.

• Matching a survival exponent, or even a leading survival constant, does not equate finite moments. Our double-Pareto density includes the required factor $1/2$ on each side; its normalization and variance convention are explicit.

• A limit $\mathbb E[X\mid X>u]/u=1$ does not establish a positive exponential moment. For survival $e^{-\sqrt u}$, direct integration gives $\int_u^\infty e^{-\sqrt t}\,dt=2(\sqrt u+1)e^{-\sqrt u}$. Dividing by survival gives $e(u)=2\sqrt u+2$, so the conditional-mean ratio tends to one despite subexponentiality.

• Maximum entropy is relative to a support, reference measure, and specified constraints. The appendix’s escort-mean calculation requires positive normalization and an admissible entropic index; stationarity alone is not a proof of a global optimum for a nonlinear constraint.

Notation follows the shared canon. The source’s lower cutoff $L$ maps to $x_m$; $L(x)$ here is a slowly varying function. We reserve $\alpha>0$ for a survival exponent, use $\nu$ for Student degrees of freedom, and use local $q$ for the entropic index. The formula $\xi=1/\alpha$ is restricted to the positive regularly varying case.

Reproduce the worked models¶

Use pareto_truncated_moment, q_exponential_logsf.

from xmath.incerto.power_laws import pareto_truncated_moment, q_exponential_logsf

moments = pareto_truncated_moment([10, 100], alpha=1.5, order=1.5)
print("Boundary moment: [" + ", ".join(f"{v:.4g}" for v in moments) + "]")
for q in (0.75, 1, 1.25):
    values = q_exponential_logsf([0, 1, 2], q, beta=2)
    print(f"q={q:.4g}: [" + ", ".join(f"{v:.4g}" for v in values) + "]")

Output

Boundary moment: [3.454, 6.908]
q=0.75: [-0, -3.466, -inf]
q=1: [-0, -2, -4]
q=1.25: [-0, -1.216, -2.079]

A log survival of -inf at the compact endpoint means zero survival.

Figure regeneration: uv run python demos/incerto/power_laws.py after installing the repository dependencies. It produces four figures: tail diagnostics, products and moment integrals, Student/double-Pareto matching, and entropy constraints. The grids and one-dimensional quadrature are bounded; no random draws, external data, fitted parameters, censoring, or sampling intervals are used. The concept-page captions give model parameters and numerical limits.

The maximum-entropy example keeps $\mathbb E\log(X/x_m)=1/2$ fixed and compares an exact Pareto law with a log-gamma alternative. The entropy gap is computed independently from the defining integrals. This checks a worked case of the analytic proof, not a general numerical search over densities.

References¶

• Taleb (2025), ch. 5, pp. 91–105; appendix B, pp. 127–133, at the section locators above. Source context and notation only where qualified.

• The regular-variation, Karamata, and subexponential notes give theorem-level references for their cited general results.