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

{{citation

| last1 = Ledoux | first1 = Michel

| last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand

| doi = 10.1007/978-3-642-20212-4

| isbn = 3-540-52013-9

| mr = 1102015

| page = 237

| publisher = Springer-Verlag | location = Berlin

| series = Ergebnisse der Mathematik und ihrer Grenzgebiete

| title = Probability in Banach spaces: Isoperimetry and processes

| url = https://books.google.com/books?id=fuclBQAAQBAJ&pg=PA237

| volume = 23

| year = 1991}}.

{{citation

| last = Pisier | first = Gilles

| doi = 10.1017/CBO9780511662454

| isbn = 0-521-36465-5

| mr = 1036275

| pages = 42–43

| publisher = Cambridge University Press, Cambridge

| series = Cambridge Tracts in Mathematics

| title = The volume of convex bodies and Banach space geometry

| url = https://books.google.com/books?id=FBRAOfpX1KEC&pg=PA42

| volume = 94

| year = 1989}}.

Category:Dimension

Category:Statistical randomness