Iso-Density Tail Geometry

Independent coordinates¶

Iso-density tail geometry studies the level sets of the iid joint density

near a large-deviation constraint such as $x_1+x_2=s$. The diagnostic asks whether the most likely configurations along the constraint sit near the diagonal, where both coordinates are moderately large, or near the axes, where one coordinate is large and the other is ordinary.

Here $X_1,X_2$ are independent copies of a continuous random variable with density $f$, and $x_1,x_2$ are its coordinate values. The sum level $s$ is fixed for each comparison. See Shared notation.

Normal versus Cauchy geometry¶

For two iid standard normal variables, the joint density is proportional to

On the line $x_1+x_2=s$,

so the joint density is maximized at the equal split $(s/2,s/2)$. A large Gaussian sum is geometrically a many-moderate-deviation event.

For two iid standard Cauchy variables,

Compare the equal split $(s/2,s/2)$ with an axial split $(0,s)$. The axial to equal density ratio is

Along a large-sum line, the Cauchy geometry increasingly favors one large coordinate rather than two equal coordinates. This is the density-contour version of the one-big-jump intuition in Subexponentiality and the Survival Tail Ratio.

Static density comparison¶

For the same sum level $s$, the normal and Cauchy axial-to-equal ratios are

The table evaluates exact density ratios, without estimating probabilities. Normal equal splits dominate axial splits more strongly as $s$ increases; Cauchy axial splits eventually dominate equal splits.

The axial points are comparison points, not the exact Cauchy maximizers for finite positive $s$. To locate those maximizers, minimize the denominator $D(x)=(1+x^2)(1+(s-x)^2)$ along the line. Its derivative factors as

For $s>2$, the two minima occur at

so the maximizing pairs approach $(1/s,s-1/s)$ and its swap as $s$ grows. They sit near the axes, while the equal split becomes a local density minimum. For $0\le s\le2$, the equal split maximizes the constrained density.

This algebra describes density along a constraint of probability zero. It does not prove a tail probability asymptotic. The Cauchy example is two-sided; applying the nonnegative-summand subexponential theorem to it requires a separate right-tail argument.

The same Cauchy marginals, a different joint law¶

The preceding calculation proves a comparison under independence. A spherical bivariate Cauchy has instead the normalized density

The multivariate Student construction derives its normalization and Cauchy marginals at $\nu=1$. Its density is strictly decreasing in squared radius. The same completed-square identity used for the Gaussian therefore proves that, on $x_1+x_2=s$, its density is maximized at the equal split for every $s$. Heavy marginal tails alone do not determine the constrained density geometry.

There is a probability comparison as well. In polar coordinates, integrating $g$ over the full circle gives

Angle is uniform and independent of radius because the joint polar density factorizes. A sector of angle width $\delta\in[0,2\pi]$, outside that circle, has probability $\delta/(2\pi\sqrt{1+u^2})$. At $u=3$ and $\delta=\pi/3$, this is $1/(6\sqrt{10})\approx0.052705$. Unlike a point-density ratio, this quantity integrates over a region of positive area. It is not a large-sum probability.

Switching correlation can destroy ellipticality¶

Consider an equal mixture of two centered bivariate normals with unit marginal variances and correlations $r$ and $-r$, where $0<r<1$. Its covariance is $I_2$, and conditioning gives characteristic function

We prove that this mixture is not elliptical. For a nondegenerate elliptical law with finite covariance, covariance is a positive scalar multiple of scatter: whitening gives a rotationally invariant law, whose covariance commutes with every orthogonal matrix and hence is scalar. Thus an elliptical law with covariance $I_2$ must be spherical. But the displayed characteristic function at $(v,0)$ is $e^{-v^2/2}$, whereas at $(v/\sqrt2,v/\sqrt2)$ it is $e^{-v^2/2}\cosh(rv^2/2)$. For $v\ne0$ these differ at the same radius. This disproves sphericality and hence ellipticality. Both marginal laws remain standard normal.

The iid Cauchy vector gives another distinction: every linear combination is Cauchy, with scale $|a|+|b|$ for coefficients $a,b$ not both zero, since its characteristic function is $e^{-|t_1|-|t_2|}$. It is a stable vector but not elliptical. To see the latter without moments, its characteristic function on either axis would force the elliptical profile to be exponential in the square root of its argument. It is also unchanged when one coordinate changes sign, forcing off-diagonal scatter zero. The equal marginal scales would then force circular level sets; $|t_1|+|t_2|$ instead has diamond level sets. Thus closure under linear combinations does not require ellipticality.

Use bivariate_student.

import numpy as np
from xmath.incerto.multivariate import bivariate_student

rng = np.random.default_rng(20261020)
sample = bivariate_student(rng, nu=1, rho=0, size=10000)
radius = np.linalg.norm(sample, axis=1)
angle = np.arctan2(sample[:, 1], sample[:, 0])
sector = (radius > 3) & (angle >= 0) & (angle <= np.pi / 3)
print(f"Sector frequency: {sector.mean():.4g}")
print(f"Exact probability: {1 / (6 * np.sqrt(10)):.4g}")

Output

Sector frequency: 0.0533
Exact probability: 0.0527

The 10,000 seeded draws illustrate an integrated probability, not a density contour level.

Figure regeneration: uv run python demos/incerto/multivariate_tails.py for the contours, the sector integral and the earlier constrained-density ratios. The chapter 6 guide links the sample-correlation and information calculations.

Caveats¶

• Iso-density geometry requires a density. Discrete, singular, censored, or heavily rounded data need a different diagnostic.

• Density at a point is not probability mass. The contour picture explains local geometry; probability statements require integration over regions.

• The iid product-density formula fails under dependence. Volatility clustering, common factors, and contagion can rotate or bend the contours.

• The examples here are symmetric and centered. Skewed losses or one-sided positive variables should first be put into a problem-specific coordinate system.

References¶

• Taleb (2025), third edition (2025), arXiv:2001.10488v4, §§6.2–6.3.1, pp. 109–114: ellipticality and independent versus shared-scale Cauchy context. The constrained extrema, sector integral, switching example and stable-vector counterexample above are proved here.

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

Backlinks¶

• Depends on: Survival Tail Ratio and Subexponentiality.

• Related diagnostic: Max-to-Sum Ratio.

• Related density geometry: Body, Shoulders, and Tails.