Concentration dimension

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

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

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 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).

References

• Ledoux, Michel; Talagrand, Michel (1991), Probability in Banach spaces: Isoperimetry and processes, Ergebnisse der Mathematik und ihrer Grenzgebiete 23, Berlin: Springer-Verlag, p. 237, doi:10.1007/978-3-642-20212-4, ISBN 3-540-52013-9, MR 1102015.
• Pisier, Gilles (1989), The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics 94, Cambridge University Press, Cambridge, pp. 42–43, doi:10.1017/CBO9780511662454, ISBN 0-521-36465-5, MR 1036275.