The Diagonal Gaussian Random Series in a Hilbert Space: Almost Sure Convergence, Finite-Dimensional Projections, and the Unique Measure with Diagonal Gaussian Projections

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with the coordinate maps $p_{n}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates and push-forwards as in Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward, let $c$ be a variance sequence, with truncations $c^{(n)}$ and diagonal Gaussian measures $\gamma_{c^{(n)}}$ on $\mathbb{R}^{n}$ ($n\in\mathbb{N}$), and let $\sqrt{s}$ be the nonnegative square root of a nonnegative real number $s$.

Claims 1 to 3 concern a probability space $(\Omega,\mathcal{F},P)$ and an independent sequence $(\xi_{k})_{k\in\mathbb{N}}$ of standard normal random variables on it. For $n\in\mathbb{N}$ let $Y_{n}:\Omega\to X$ be the map $Y_{n}(\omega)=\sum_{k=1}^{n}\sqrt{c_{k}}\,\xi_{k}(\omega)\,e_{k}$, and let $G$ be the set of those $\omega\in\Omega$ for which the series $\sum_{k=1}^{\infty}c_{k}\,\xi_{k}(\omega)^{2}$ converges. Define $Y:\Omega\to X$ (well defined by claim 1) by $Y(\omega)=\lim_{n\to\infty}Y_{n}(\omega)$, the limit in $(X,d)$, for $\omega\in G$, and $Y(\omega)=0_{X}$ for $\omega\notin G$.

1. (Almost sure convergence)¶ $G\in\mathcal{F}$, $P(G)=1$, and for every $\omega\in G$ the sequence $(Y_{n}(\omega))_{n\in\mathbb{N}}$ converges in $(X,d)$.

2. (A random element)¶ $Y$ is a random element of $(X,d)$, and $\langle Y(\omega),e_{k}\rangle=\sqrt{c_{k}}\,\xi_{k}(\omega)$ for every $\omega\in G$ and every $k\in\mathbb{N}$.

3. (Projections)¶ The law $P_{Y}$ of $Y$ belongs to $\mathcal{P}(X)$, and $(p_{n})_{\#}P_{Y}=\gamma_{c^{(n)}}$ for every $n\in\mathbb{N}$.

4. (The unique measure with these projections)¶ There is exactly one $\mu\in\mathcal{P}(X)$ such that $(p_{n})_{\#}\mu=\gamma_{c^{(n)}}$ for every $n\in\mathbb{N}$. For every probability space and every independent sequence of standard normal random variables on it, this $\mu$ is the law $P_{Y}$ of the corresponding map $Y$ defined above.