Gaussian Valued Random Variables in Geometry of Banach Spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:26:49Z http://mathoverflow.net/feeds/question/114448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114448/gaussian-valued-random-variables-in-geometry-of-banach-spaces Gaussian Valued Random Variables in Geometry of Banach Spaces Nirman 2012-11-25T20:29:05Z 2012-11-26T10:33:54Z <p>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). </p> <p>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).</p> http://mathoverflow.net/questions/114448/gaussian-valued-random-variables-in-geometry-of-banach-spaces/114510#114510 Answer by R W for Gaussian Valued Random Variables in Geometry of Banach Spaces R W 2012-11-26T10:33:54Z 2012-11-26T10:33:54Z <p>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). </p> <p>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.</p>