Chapter 6 — Thick tails in higher dimensions

A tail exponent describes one coordinate. It does not tell us whether large coordinates arrive together, whether a covariance exists, or what a matrix of observations can estimate. Chapter 6 adds those questions. Begin with density geometry, then use the multivariate Student construction to separate shared scale, independence, and correlation.

This guide uses Taleb (2025), third edition (2025), arXiv:2001.10488v4, chapter 6, printed pp. 107–120, and attached appendix A, pp. 121–125. The substantive numbered sections end on p. 119; p. 120 is the chapter transition. The notes supply arguments and examples, with proved and cited results identified locally.

Reading map¶

Three meanings of zero¶

For a bivariate Student with degrees of freedom $\nu>0$ and scatter $\Sigma=\left(\begin{smallmatrix}1&\rho\\\rho&1\end{smallmatrix}\right)$, setting $\rho=0$ means the scatter axes are orthogonal. If $\nu>2$, Pearson correlation also equals zero. Independence still fails for every finite $\nu$, because the two coordinates share the scale.

At $\nu=1$ this is the spherical Cauchy. Its Pearson coefficient is undefined but its mutual information is finite:

The entropy calculation proves the value. Independent Cauchy coordinates instead have mutual information zero and nonelliptical contours. Thus a marginal family and one matrix parameter do not specify the dependence model.

Qualifications to carry into the source¶

• A Student scatter matrix is $\Sigma$, while covariance, when finite, is $\nu\Sigma/(\nu-2)$. The density needs its normalizing constant. The mixture proof keeps both distinctions explicit.

• Stable vectors need not be elliptical. The independent Cauchy counterexample has stable linear projections and nonelliptical level sets. A singular law need not have a density, but stability alone does not imply its absence.

• A semicircle statement needs a matrix ensemble and scaling. A finite fourth moment is not necessary for the cited empirical spectral limit; extreme eigenvalues and sample covariance spectra are separate questions.

• The uncentered cosine in §6.6 differs from centered sample Pearson correlation. Neither finite statistic creates a missing population moment.

• For $V=b+aX+\varepsilon$ with independent, centered finite-variance error and standard normal $X$, the population target is $a^2/(a^2+\operatorname{Var}\varepsilon)$. The proof gives consistency but no universal upward-bias or $O(n^{-2})$ rate. An undefined infinite-variance target must not be assigned zero by a ratio of infinities. The source’s historical regressions are not reproduced.

• An absorbing state is eventually reached in our homogeneous model only when its conditional failure probability is positive. Irreversibility without persistent risk is insufficient, as the appendix example shows.

These qualifications bound the teaching claims. They do not assess portfolio performance, historical data quality, or the source’s broad prescriptions. Notation follows the shared canon: $\alpha$ denotes a survival exponent, while $\nu$ denotes Student degrees of freedom (equal to each marginal’s exponent here). $Y$ is mixing variance, $G=1/Y$ precision, $V$ a regression response, and $T$ a transition matrix. The appendix’s state probabilities must not be read as point densities.

Worked cases¶

Use student_mutual_information, regression_population_r2, symmetric_spectrum.

from xmath.incerto.multivariate import (
    student_mutual_information, regression_population_r2, symmetric_spectrum,
)

print(f"Cauchy information: {student_mutual_information(1, rho=0):.4g}")
print(f"Regression target: {regression_population_r2(1, error_variance=3):.4g}")
values = symmetric_spectrum([[0, 2], [2, 0]])
print("Matrix eigenvalues: [" + ", ".join(f"{v:.4g}" for v in values) + "]")

Output

Cauchy information: 0.2242
Regression target: 0.25
Matrix eigenvalues: [-2, 2]

Information is measured in nats; the finite matrix example implies no random-matrix limit.

Figure regeneration: uv run python demos/incerto/multivariate_tails.py. Five single-column figures use exact density/entropy formulas, checked one-dimensional quadrature, and bounded seeded simulations:

The sampling experiments use PCG64 seed 20261018, without clipping or discarding draws. Printed quantiles describe the simulated sampling laws; they do not certify an asymptotic theorem. The runner also checks a Cauchy sector probability, a finite spectral identity, and the rare-Pareto example. There is no external dataset or fitted tail model.

Continue to moment estimation for the distinction between an existing moment and a stable estimator, or return to the SCoFT overview.