I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."



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> For each $\epsilon >0$ there is a number $\eta^\prime(\epsilon) > 0$ with the following property. Let $X$ be a Gaussian r.v. with values in a Banach space $B$ of dimension $N.$ Then $B$ contains a subspace $F$ of dimension $n = [\eta^\prime(\epsilon) d(X)]$ which is $(1+\epsilon)$-isomorphic to $\ell^n\_2.$ Conversely, if $B$ contains a subspace $F$ with $F \stackrel{1+\epsilon}{\sim} \ell^n\_2,$ then there is a $B$-valued Gaussian r.v. $X$ such that $d(X) \geq (1+\epsilon)^{-2}n.$
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This statement uses the "dimension" $d(X)$ of a Gaussian variable, 
$$ d(X) = \mathbb{E}\|X\|^2/\sigma(X)^2, $$
where 
$$ \sigma(X)^2 = \sup \{ \mathbb{E} \xi(X)^2 \mid \|\xi\|\_{B^\star} \leq 1 \} $$
is the weak variance of $X.$


For this to match the usual $n = O(\log N)$ statement of the theorem, you'd need a lower bound on $d(X)$ of order $\log N,$ and as a sanity check an upper bound of $O(N).$


Any hints or references on how to show these two bounds on $d(X)$? Pisier states that the upper bound $d(X) \leq N$ is easy to show, but I've not been able to prove even that. 



Why is the dimension of Gaussian variables is bounded by the dimension of the space?
None of these estimates is trivial. The upper bound follows from John's theorem that the Banach-Mazur distance between an N dimensional normed space and an N dimensional Euclidean space is at most $\sqrt N$. The lower bound is more involved and uses the Dvoretzky-Rogers lemma. A good reference for the Gaussian approach to Dvoretzky's theorem is Pisier's book: The volume of convex bodies and Banach space geometry. For a more geometrical approach see: Milman and Schechtman: Asymptotic theory of finite-dimensional normed spaces.
The theorem and it's proof is presented in other books as well.


