Concentration dimension

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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[edit]

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

\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[edit]

  • If B is n-dimensional Euclidean space Rn with its usual Euclidean norm,[clarification needed] 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[edit]