Consider a probability space (Ω,Σμ) and the corresponding Hilbert space
$latex H=L^2_0(\Omega,\mu)$
of real-valued square-integrable functions with zero mean:
$latex \int_{\Omega}{f(x)d\mu(x)}=0\text{ for } f\in H.$
In H, we can define the subset
$latex G=\left{X\,:\, \Omega\rightarrow \mathbf{R}\,\mid\, X\text{ is a centered gaussian variable}},$
(where gaussian is in the non-strict sense: constant variables are considered gaussians, with variance 0; this has the effect in particular that G is not empty, as it contains the zero random variable).
Here’s a question I’m vaguely puzzled about: does the set G have any known interesting structure? Note that it is not a vector space, but it satisfies the following conditions:
(1) it is closed for the norm topology;
(2) it is stable under scalar multiplication (so it is geometrically a cone — already some kind of structure);
(3) if X and Y are in G, and are orthogonal [i.e., are independent; recall that all variables are centered], then the linear space spanned by X and Y is entirely contained in G;
(4) (almost) conversely, a result of Cramer states that if X and Y are in H, are orthogonal, and their sum is in G, then X and Y are in G.
These four conditions together are quite strong, and I’m wondering how far they are from capturing the truth about G. In other words, could it be that, in some sense, any subset M of H, or indeed of an arbitrary Hilbert space, satisfying (1)–(4) is necessarily related to a set of gaussian random variables? (This makes sense since all conditions are expressed completely in the language of Hilbert spaces, with no reference to the underlying probability space, once “orthogonal” has been used instead of “independent”).
Or are there interesting subsets M which are genuinely different? I don’t really have a clue (or an opinion), but I haven’t thought about it very much either…
Since there are many known characterizations of the normal distribution, these may well be questions which have already been answered, but for instance the book of Bogachev on Gaussian Measures (linked-to above) does not seem to contain anything of this kind.
