Question

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).

Answer 1

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).

As for the difference between Gaussian and Bernoulli random variables - this is a confusion based on a certain ambiguity of the probabilistic language. In fact, the qualifier "Bernoulli" can only be applied to a family of random variables (and is synonymous to i.i.d. - independent identically distributed), but not to a single random variable. If one talks about measures instead of random variables, then the difference becomes obvious - a Gaussian measure is a measure on a linear space, whereas a Bernoulli measure is a measure on a product space.