concentration of the norm of a random vector

Concentration of the norm is the first and most vivid surprise of high-dimensional geometry: a random vector with independent sub-Gaussian coordinates does not spread its length out, but instead has a Euclidean norm tightly concentrated around sqrt(n), living essentially on a thin spherical shell. This 'thin-shell phenomenon' is why high-dimensional intuition differs so radically from the low-dimensional picture, and it is the engine behind random projections, nearest-neighbour methods, and the geometry of data clouds.

Let X = (X_1, ..., X_n) have independent, mean-zero, unit-variance, sub-Gaussian coordinates with psi_2 norm at most K. Then E[||X||_2^2] = sum E[X_i^2] = n, so the typical length is sqrt(n). The concentration statement is that ||X||_2 itself (not its square) concentrates: ||X||_2 - sqrt(n) is sub-Gaussian with a variance proxy that does NOT grow with n, P(| ||X||_2 - sqrt(n) | >= t) <= 2 exp(-c t^2 / K^4) for all t >= 0. The deviation of the norm is O(1), independent of dimension — the relative fluctuation ||X||/sqrt(n) - 1 is therefore of order 1/sqrt(n) and vanishes. The proof is a clean two-step: first ||X||_2^2 = sum X_i^2 is a sum of independent sub-exponential variables (each X_i^2 has mean 1), so by Bernstein it concentrates around n with deviation O(sqrt(n)); then the inequality | ||X|| - sqrt(n) | <= |||X||^2 - n| / sqrt(n) (since a - b <= (a^2 - b^2)/b for a, b >= 0 appropriately) converts the O(sqrt(n)) deviation of the square into the O(1) deviation of the norm. The square-root is what tames the fluctuation.

The consequences are everywhere in data science. In R^n with n large, almost all the volume of a ball sits near its surface, almost all of a Gaussian's mass sits near the sphere of radius sqrt(n), and two independent high-dimensional Gaussian vectors are very nearly orthogonal with inner product of order sqrt(n) against lengths of order n. These facts underlie the Johnson-Lindenstrauss lemma and the success of random features. The honest caveat: the clean sqrt(n) concentration needs the coordinates to be sub-Gaussian (or at least to have a uniformly controlled sub-exponential X_i^2); for heavy-tailed coordinates the thin-shell can fail or widen, and the independence hypothesis matters — strongly dependent coordinates need not concentrate at all, which is exactly the content of the still-partially-open 'thin-shell conjecture' for general log-concave vectors.