In mathematics — specifically, in probability theory — the concentration dimension of a Banach space-valued random variable is a numerical measure of how "spread out" the random variable is compared to the norm on the space.

Definition

Let (B, || ||) be a Banach space and let X be a Gaussian random variable taking values in B. That is, for every linear functional ℓ in the dual space B^∗, the real-valued random variable ⟨ℓ, X⟩ has a normal distribution. Define

${\displaystyle \sigma (X)=\sup \left\{\left.{\sqrt {\operatorname {E} [\langle \ell ,X\rangle ^{2}]))\,\right|\,\ell \in B^{\ast },\|\ell \|\leq 1\right\}.}$

Then the concentration dimension d(X) of X is defined by

${\displaystyle d(X)={\frac {\operatorname {E} [\|X\|^{2}]}{\sigma (X)^{2))}.}$

Examples

• If B is n-dimensional Euclidean space Rⁿ with its usual Euclidean norm, and X is a standard Gaussian random variable, then σ(X) = 1 and E[||X||²] = n, so d(X) = n.
• If B is Rⁿ with the supremum norm, then σ(X) = 1 but E[||X||²] (and hence d(X)) is of the order of log(n).