Strong and Weak Convergence of Noise Fields Along Couplings of Vanishing Noise Cost

In the setting of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, let $\mathcal{P}^{a}_{\rho}$ be the set of The Measures Noise-Connected to the Reference Measure §space, let $\nu\in\mathcal{P}^{a}_{\rho}$ and let $(\nu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{a}_{\rho}$. Limits of sequences of real numbers are those of that definition. For $\sigma\in\mathcal{P}(X)$, $L^{2}(\sigma;X^{a})$ is the space of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, with inner product $\langle\cdot,\cdot\rangle_{\sigma}$. For $n\in\mathbb{N}$, $\Pi^{a}(\nu_{n},\nu)$ is the set of Couplings of Finite Noise Cost and Their Noise Cost §couplings, contained in the set $\Pi(\nu_{n},\nu)$ of couplings of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background by that definition, and for $\pi\in\Pi^{a}(\nu_{n},\nu)$ the noise cost $I^{a}(\pi)$ is a nonnegative real number by Couplings of Finite Noise Cost and Their Noise Cost §cost. For $\pi\in\Pi(\nu_{n},\nu)$, $q'\in L^{2}(\nu_{n};X^{a})$ and $\eta\in L^{2}(\nu;X^{a})$, the discrepancy $\int_{X\times X}|q'(x)-\eta(y)|_{a}^{2}\,\pi(dz)$ is the real number of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy and the cross pairing $\mathcal{K}^{a}(q',\eta,\pi)$ that of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, both applied with $\nu_{n}$ and $\nu$ in place of its $\nu$ and $\mu$.

1. (Couplings of vanishing noise cost)¶ A sequence $(\pi_{n})_{n\in\mathbb{N}}$ with $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$ for every $n\in\mathbb{N}$ is a sequence of couplings of vanishing noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$ if $\lim_{n\to\infty}I^{a}(\pi_{n})=0$.

2. (Strong convergence along couplings)¶ Let $(\pi_{n})_{n\in\mathbb{N}}$ be a sequence of couplings of vanishing noise cost from $(\nu_{n})_{n\in\mathbb{N}}$ to $\nu$, let $q_{n}\in L^{2}(\nu_{n};X^{a})$ for every $n\in\mathbb{N}$, and let $q\in L^{2}(\nu;X^{a})$. The sequence $(q_{n})_{n\in\mathbb{N}}$ converges strongly to $q$ along $(\pi_{n})_{n\in\mathbb{N}}$ if

$$\lim_{n\to\infty}\int_{X\times X}|q_{n}(x)-q(y)|_{a}^{2}\,\pi_{n}(dz)=0 .$$

3. (Weak convergence along couplings)¶ With $(\pi_{n})_{n\in\mathbb{N}}$, $(q_{n})_{n\in\mathbb{N}}$ and $q$ as in clause 2, the sequence $(q_{n})_{n\in\mathbb{N}}$ converges weakly to $q$ along $(\pi_{n})_{n\in\mathbb{N}}$ if

$$\lim_{n\to\infty}\mathcal{K}^{a}(q_{n},\eta,\pi_{n})=\langle q,\eta\rangle_{\nu}\qquad\text{for every }\eta\in L^{2}(\nu;X^{a}).$$