Vector spaces of random variables having zero expectation - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:43:57Z http://mathoverflow.net/feeds/question/45275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45275/vector-spaces-of-random-variables-having-zero-expectation Vector spaces of random variables having zero expectation Olumide 2010-11-08T07:10:33Z 2010-11-08T11:18:58Z <p>Edit: Robin's comments appear to have made the matter a lot clearer to me. I now suppose that the vector space of random variables with zero expectation are studied in the context of second order stationary processes.</p> <p>The other question remains: are vector spaces of random variables with non-zero expectation also studied?</p> http://mathoverflow.net/questions/45275/vector-spaces-of-random-variables-having-zero-expectation/45282#45282 Answer by Shai Covo for Vector spaces of random variables having zero expectation Shai Covo 2010-11-08T09:49:58Z 2010-11-08T09:49:58Z <p>Suppose we are in the context of second order stationary processes. Then, quoting from Wikipedia (entry on Stationary process), "such a process will be wide sense stationary if the mean and correlation functions are finite". In turn (again, see Wikipedia), the mean function $m_x (t) = {\rm E}{ x(t)}$ of a wide-sense stationary process must be constant. This can account for the assumption of zero expectation you indicated. The situation is similar with regard to Gaussian processes, where it is "frequently assumed" that the process has zero mean; then, as is well known, the law of the process is determined by its covariance.</p> http://mathoverflow.net/questions/45275/vector-spaces-of-random-variables-having-zero-expectation/45285#45285 Answer by zhoraster for Vector spaces of random variables having zero expectation zhoraster 2010-11-08T10:11:49Z 2010-11-08T10:11:49Z <p>One can, of course, think of $L^2(\Omega)$ as of a Hilbert space with a scalar product $E[\xi\eta]$. But for random variables much more important is the covariance $E[\xi\eta]-E[\xi]E[\eta]$. Though it looks at first sight as a scalar product, unfortunately it's not, as $\mathrm{cov}(\xi,\xi)=0$ doed not imply $\xi=0$. However, on the space of centered r.v.'s it is a scalar product. And this Hilbertian structure fully determines the laws in some cases, like a Gaussian case, as Shvai Covo already mentioned. And also this Hilbertian structure plays a very important role for (weakly) stationary processes (also noted by Shvai Covo).</p> <p>Vector spaces of non-centered random variables are not so popular. One of applications which I think about is financial mathematics, though there you more often work with some cones rather than full vector spaces. Still, a lot of machinery is based (especially in discrete time) on some vector space techniques.</p>