Replacing the Tail of a Probability Measure on a Hilbert Space by an Independent Diagonal Gaussian Tail: Same Head and Convergence in the Quadratic Wasserstein Distance

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, with $p_{n}$ and $Q_{n}$ as in 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 diagonal Gaussian measure $\gamma_{c}$, let $\mu\in\mathcal{P}_{2}(X)$, the set of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, and let $W_{2}$ be the quadratic Wasserstein distance. For $n\in\mathbb{N}$ let $\mathbb{R}^{n}\times X$ carry the product metric, whose Borel $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ by The Borel Sigma-Algebra of a Product of Two Separable Metric Spaces is the Product Sigma-Algebra §product; let $\Phi_{n}:\mathbb{R}^{n}\times X\to X$, $\Phi_{n}(u,w)=p_{n}^{*}(u)+w$, be the Borel map of Head and Tail of a Diagonal Gaussian Measure on a Hilbert Space are Independent §synthesis; let $(p_{n})_{\#}\mu\otimes(Q_{n})_{\#}\gamma_{c}$ be the probability measure on $\mathcal{B}(\mathbb{R}^{n})\otimes\mathcal{B}(X)$ given by Existence and Uniqueness of the Product Measure; and let $\mu^{(n)}$ be its image measure under $\Phi_{n}$, a Borel probability measure on $X$ by claim 1 of Image Measures, Measures with Densities, and Change of Variables:

$$\mu^{(n)}=(\Phi_{n})_{\#}\bigl((p_{n})_{\#}\mu\otimes(Q_{n})_{\#}\gamma_{c}\bigr).$$

1. (Same head)¶ For every $n\in\mathbb{N}$, $\mu^{(n)}\in\mathcal{P}_{2}(X)$ and $(p_{n})_{\#}\mu^{(n)}=(p_{n})_{\#}\mu$.

2. (Convergence)¶ The sequence $\bigl(W_{2}(\mu^{(n)},\mu)\bigr)_{n\in\mathbb{N}}$, defined by claim 1, converges to $0$, and $\mu^{(n)}\Rightarrow\mu$ in the sense of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §weak.