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
![{\displaystyle \sigma (X)=\sup \left\{\left.{\sqrt {\operatorname {E} [\langle \ell ,X\rangle ^{2}]}}\,\right|\,\ell \in B^{\ast },\|\ell \|\leq 1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb6bd4ab984d75d911ef406f55c3a5c68dc808d)
Then the concentration dimension d(X) of X is defined by
![{\displaystyle d(X)={\frac {\operatorname {E} [\|X\|^{2}]}{\sigma (X)^{2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab4a629db1df1074f145ba714742a4c68abe051a)
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, 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.