# Construction of Gaussian Hilbert spaces

I am reading the very first chapter of "Gaussian Hilbert Spaces" by S. Janson.

Definition: A Gaussian Hilbert space is a closed subspace of $L^2(\Omega, \mathcal{F}, P)$ consisting of centered Gaussian random variables.

Example 1: Let $\xi$ be any non-degenerate, normal variable with mean zero. Then $\{ t \xi: t \in \mathbf{R} \}$ is a one-dimensional Gaussian Hilbert space.

Example 2: Let $\xi_1, \dots, \xi_n$ have a joint normal distribution with mean zero. Then their linear span $\{ \sum_{i = 1}^n t_i \xi_i : t_i \in \mathbf{R} \}$ is a finite-dimensional Gaussian Hilbert space.

I am missing something really fundamental and, presumably, trivial for other people. In both cases, the spaces are composed of one-dimensional Gaussian random variables with zero means and all possible finite variances. Then the question is: what Gaussian random variable cannot be obtained using the first approach while it can easily be constructed using the second one? What is the difference? Or may be it is not about individual variables but rather about their relationships with each other. In the first case, all the variables are perfectly correlated whereas the second space is much reacher in this regard, especially, if we assume that $\xi_1, \dots, \xi_n$ are independent.

Thank you!

Regards, Ivan

## 1 Answer

The difference lies in the dimension of the Hilbert space generated. Take $\{ \xi_1, \xi_2\} \sim N(0, I_2)$, which form an orthonormal basis of a two dimensional subspace. This is clearly not the same as the one dimensional subspace generated by $\xi_1$.

Indeed you seem to be missing something fundamental: the distribution of a random variable $\xi$, or equivalently, its push-forward measure on $\mathbb{R}$, does not characterize $\xi$. For example, if $\xi \sim U[0,1]$, then $-\xi +1 \sim U[0,1]$.

• Thank you for the answer. Can you please elaborate a bit more on "does not characterize $\xi$?" You mean the way it interacts with the other variables in the space (the joint distribution)? For me the difference becomes more apparent when we take two variables and observe that their correlation is nonzero. And I am not sure if your last example is correct. – Ivan Dec 18 '13 at 10:14
• Should have been $1 - \xi$. The point was that random variables are just (measurable) functions on a underlying probability space. It's easy to cook up different functions with the same distribution. – Michael Dec 18 '13 at 18:22