undefined — math.log

The standard story

You have $n$ i.i.d. random variables $X_1, \ldots, X_n$ with mean $\mu$ and variance $\sigma^2$ . Two classical questions:

The CLT question: what does the average $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ look like for large $n$ ? After centering and scaling:

$\frac{\sqrt{n}(\bar{X}_n - \mu)}{\sigma} \xrightarrow{d} \mathcal{N}(0,1)$

The EVT question: what does the maximum $M_n = \max(X_1, \ldots, X_n)$ look like for large $n$ ? For a broad class of light-tailed distributions, after appropriate centering $b_n$ and scaling $a_n$ :

$\frac{M_n - b_n}{a_n} \xrightarrow{d} \text{Gumbel}$

where the Gumbel distribution has CDF $F(x) = e^{-e^{-x}}$ . These look like unrelated facts. But they aren’t.

Think in high dimensions

Here is a reframing that changes how both results feel. Consider the random vector $\mathbf{X} = (X_1, \ldots, X_n) \in \mathbb{R}^n$ as a single point. The CLT and EVT are both statements about where this point lives in $\mathbb{R}^n$ — they just measure distance differently.

The CLT cares about the $\ell^2$ norm — the Euclidean distance from the origin:

$\|\mathbf{X}\|_2 = \sqrt{X_1^2 + \cdots + X_n^2}$

EVT cares about the $\ell^\infty$ norm — the maximum coordinate:

$\|\mathbf{X}\|_\infty = \max(|X_1|, \ldots, |X_n|)$

The $\ell^2$ unit ball is the sphere. The $\ell^\infty$ unit ball is the cube $[-1, 1]^n$ . The two limit theorems are concentration phenomena on these two different geometric objects.

The CLT as thin-shell concentration

Suppose each $X_i \sim \text{Uniform}(-1, 1)$ . The point $\mathbf{X}$ lives in the cube $[-1,1]^n$ . Where exactly? By the CLT, since each $X_i^2$ has mean $m_2 = \frac{1}{3}$ , the law of large numbers gives:

$\frac{\|\mathbf{X}\|_2^2}{n} = \frac{1}{n}\sum_{i=1}^n X_i^2 \to m_2 = \frac{1}{3}$

So $\|\mathbf{X}\|_2 \approx \sqrt{n/3}$ with high probability. The point doesn’t fill the cube — it concentrates on a thin spherical shell at radius $\sqrt{nm_2}$ , deep inside the cube. This is the thin-shell phenomenon, and the CLT for the coordinates is the one-dimensional shadow of this concentration.

EVT as face concentration

Now ask where $\mathbf{X}$ sits relative to the cube’s boundary. The $\ell^\infty$ distance to the boundary of $[-1,1]^n$ is $1 - \|\mathbf{X}\|_\infty$ . For large $n$ , this distance is tiny — the point is very close to the surface.

More precisely, the maximum $M_n = \|\mathbf{X}\|_\infty$ satisfies:

$P\!\left(n(1 - M_n) \leq t\right) \to 1 - e^{-t}$

which is the exponential distribution — a Gumbel shifted to have location 0. Setting $M_n = 1 - t/n + o(1/n)$ and zooming in, you see the full Gumbel limit. The point $\mathbf{X}$ is concentrating on the face of the cube, not on the spherical shell inside it.

The CLT is $\ell^2$ concentration: mass moves to a sphere inside the cube. EVT is $\ell^\infty$ concentration: mass moves to the surface of the cube. They are dual directions of the same phenomenon.

Visualizing the radial distribution

The code below computes the distribution of $\|\mathbf{X}\|_2$ (left: CLT shell) and $\|\mathbf{X}\|_\infty$ (right: EVT face) for a random point in the $n$ -cube, and plots both. Try increasing $n$ to watch the concentration sharpen.

Cube concentration: CLT vs EVT ·

import numpy as np
import matplotlib.pyplot as plt

n = 100         # dimension — try 50, 200, 500
N = 20000       # samples

rng = np.random.default_rng(42)
X = rng.uniform(-1, 1, size=(N, n))

l2   = np.linalg.norm(X, axis=1)      # ell^2 norms
linf = np.max(np.abs(X), axis=1)      # ell^inf norms

fig, axes = plt.subplots(1, 2, figsize=(11, 4))

# CLT: ell^2 norm concentrates at sqrt(n/3)
expected_l2 = np.sqrt(n / 3)
axes[0].hist(l2, bins=80, density=True, color='#c8a96e', alpha=0.85, edgecolor='none')
axes[0].axvline(expected_l2, color='#e06c6c', lw=1.5, linestyle='--',
                label=f'sqrt(n/3) = {expected_l2:.2f}')
axes[0].set_title(f'ell^2 norm  (n={n})')
axes[0].set_xlabel('||X||_2')
axes[0].legend(fontsize=9)

# EVT: ell^inf norm concentrates near 1 (cube face)
axes[1].hist(linf, bins=80, density=True, color='#6e9ec8', alpha=0.85, edgecolor='none')
axes[1].axvline(1.0, color='#e06c6c', lw=1.5, linestyle='--', label='cube face (= 1)')
axes[1].set_title(f'ell^inf norm  (n={n})')
axes[1].set_xlabel('||X||_inf')
axes[1].legend(fontsize=9)

plt.tight_layout()
plt.show()
print(f"Mean ||X||_2  = {l2.mean():.4f}  (expected sqrt(n/3) = {expected_l2:.4f})")
print(f"Mean ||X||_inf = {linf.mean():.6f}  (concentrates to 1 as n grows)")

The Gumbel limit explicitly

Let’s verify the Gumbel convergence numerically. After standardizing $M_n$ by its median and IQR, the distribution should approach the standard Gumbel $F(x) = e^{-e^{-x}}$ .

Gumbel convergence ·

Why this matters for jump detection

In high-frequency financial data, we observe a process at times $0, \Delta_n, 2\Delta_n, \ldots, 1$ with $\Delta_n \to 0$ . Each increment $\Delta_i X = X_{i\Delta_n} - X_{(i-1)\Delta_n}$ is roughly normal under a pure diffusion. The vector of increments lives in $\mathbb{R}^n$ .

Under no jumps, the CLT governs the bulk: the realized volatility $\sum (\Delta_i X)^2$ concentrates at $\sigma^2$ , the $\ell^2$ shell picture. A jump test asks whether the maximum increment $\max_i |\Delta_i X|$ is consistent with the diffusion, or whether it sits too far from the expected Gumbel quantile — the $\ell^\infty$ picture.

The same geometric duality that makes CLT and EVT seem like separate subjects is exactly what makes them complementary tools for this problem: you need both to see the full picture of a semimartingale.

References & further reading

The thin-shell concentration phenomenon is developed rigorously in asymptotic geometric analysis — see Klartag (2007) and the survey by Guedon & Milman. For the EVT side, the norm-based picture appears in multivariate extreme value theory via the exponent measure; see de Haan & Ferreira (2006), Chapter 6. For jump detection in the semimartingale setting specifically, see Bibinger (2024) and Jacod & Protter, Discretization of Processes.