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

$\sigma (X)=\sup \{\sqrt{\mathrm{E}[⟨\ell ,X{⟩}^2]}|\ell \in B^{\ast },\Vert \ell \Vert \le 1\}.$

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

$d(X)=\frac{\mathrm{E}[\Vert X{\Vert }^2]}{\sigma (X{)}^2}.$

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 (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, https://books.google.com/books?id=fuclBQAAQBAJ&pg=PA237 .
• 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, https://books.google.com/books?id=FBRAOfpX1KEC&pg=PA42 .