# 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).

-

## 1 Answer

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.

-
Isn't Bernoulli usually IID 0-1 valued (in which case, perhaps the OP means Steinhaus, i.e. IID plus-minus 1) –  Yemon Choi Nov 26 '12 at 10:36
Not necessarily - the usage varies. Sometimes people go as far as to distinguish between a "Bernoulli process" and a "Bernoulli scheme" (the values being 0,1 in the first case and arbitrary in the second case). Also sometimes one calls "Bernoulli" a random variable whose distribution is concentrated on 0,1. But I am pretty sure this is not what was meant by the OP. –  R W Nov 26 '12 at 14:15
Thanks. That definitely helps! –  Nirman Nov 26 '12 at 16:16