For infinite-dimensional Gaussian measures, we often see the definition of Gaussian random variables like this:
Let $H(\Omega;\mathbb{R})$ be a separable Hilbert space. A random
variable $u \in H$ is said to be a Gaussian random variable if
$\langle u,v \rangle$ is Gaussian (i.e., $\langle u,v \rangle$ follows
the normal distribution on $\mathbb{R}$) for all $v \in H$.
Even more generally, I found in this lecture note (Definition 2.1) that Gaussian r.v. in topological vector space is defined as follows
Let $W$ be a topological vector space, and $\mu$ a Borel probability
measure on $W$. $\mu$ is Gaussian iff, for each continuous linear
functional $f\in W_*$, the pushforward $\mu \circ f^{−1}$ is a Gaussian
measure on $\mathbb{R}$.
My question might be a bit vague: I would like to know the intuition and motivation of such definition. It seems to me that the infinite-dimensional Gaussian measure is defined via its finite-dimensional analogy. I can think of the analogy from characteristic function $\mathbb{E}[e^{i \langle u,v \rangle}]$ or that linear combinations of Gaussians $\Leftrightarrow$ Gaussian. What benefits does this kind of definition bring? Are there other definitions of Gaussian r.v.s in infinite-dimensional spaces?
Found a similar question from searching.
Best Answer
Even in finite dimensions this definition is more convenient since it is independent of coordinates. If you are interested in geometric applications this is what you need.
This definition has the advantage that clarifies the nature of the the various invariants. Here are some more details.
A Gaussian measure on a a topological vector space is a Borel measure $\mu$ such that any continuous linear functional $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bE}{\mathbb{E}}$$\newcommand{\bC}{\mathbb{C}}$ $\xi: V\to\bR$, viewed as a random variable has a Gaussian distribution. The Gaussian measures on $\bR$ are characterized by their mean and variance. Denote by $m(\xi)$ and respectively $V(\xi)$ the mean and respectively variance of $\xi$. Observe that the map $$ m:V^*\to\bR,\;\; \xi \mapsto m(\xi)\in \bR $$ is linear so it naturally lives in $V^{**}$, the bidual of $V$. With a bit of luck $m$ lands in $V\subset V^{**}$. This explains why in many applications some form of reflexivity is assumed about $V$.
As for the variance $V(-)$, note that it is defined by the covariance form $$ C: V^*\times V^*\to \bR,\;\;C(\xi,\eta)=\bE\big[\big(\,\xi-m(\xi)\,\big)\big(\,\eta-m(\eta)\,\big)\big]. $$ The covariance form is a symmetric nonnegative definite form again on the dual $V^*$ and $V(\xi)=C(\xi,\xi)$.
The Fourier transform (a.k.a. the characteristic function) is then the function $\newcommand{\ii}{\boldsymbol{i}}$ $$ \widehat{\mu}:V^*\to\bC,\;\;\widehat{\mu}(\xi)= \bE_\mu\big[ e^{\ii \xi}\,\big]=\int_V e^{\ii \xi(v)} \mu[dv]. $$ The characteristic function $f_\xi$ of $\xi$ is $$ f_\xi(t)= \widehat{\mu}(t\xi),\;\;t\in\bR. $$ The book Gaussian measures by V. Bogachev adopts this point and it is worth consulting it. One source that I like very much is the fourth volume of the treatise on generalized functions by Gelfand
If $V$ is finite dimensional then for any $m\in V^{**}$ and $C$ symmetric nonnegative definite form on $V^*$ there exists a unique Gaussian measure with mean $m$ and covariance form $C$. This is no longer universally true in infinite dimensions. This a rather subtle issue. Things work out nicely if $V$ is the dual of nuclear space, for example if $V=C^{-\infty}(\bR^n)$ the dual of $C^\infty_0( \bR^n)$.