Concentration dimension

In mathematics &mdash; specifically, in probability theory &mdash; 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&lowast;, the real-valued random variable &lang;ℓ, X&rang; has a normal distribution. Define


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

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


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

Examples

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