# Gaussian Valued Random Variables in Geometry of Banach Spaces

Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very first chapter - "Dvoretzky's theorem by Gaussian Methods" there are definitions using B valued gaussian random variables X (where B is the Banach space under consideration).

Intuitively, what is the reason that would make one look toward gaussian variables - as opposed to Bernoulli rv (which I guess are also used in several definitions).

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Gaussian measures is more or less the only naturally defined'' class of measures on infinite dimensional Banach spaces. There are no translation invariant (or even quasi-invariant) measures, so that one can not define measures by their densities with respect to a canonical one (like what one does by using the Lebesgue measure in the finite dimensional case).