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.

DefinitionEdit

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

 

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

 

ExamplesEdit

  • 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).

ReferencesEdit

  • Ledoux, Michel; Talagrand, Michel (1991), Probability in Banach spaces: Isoperimetry and processes, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 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, vol. 94, Cambridge University Press, Cambridge, pp. 42–43, doi:10.1017/CBO9780511662454, ISBN 0-521-36465-5, MR 1036275.