Tail Replacement in the Noise Wasserstein Distance: a Measure of Finite Relative Entropy is Approximated by the Gaussian-Tail Extensions of Its Rescaled Heads

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, let $\kappa$ be a positive real number with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $\mu\in\mathcal{P}(X)$ have finite relative entropy with respect to $\gamma_{c}$. For $n\in\mathbb{N}$, $\tilde{\mu}_{n}$ has finite relative entropy with respect to $\tilde{\gamma}_{n}$ by Gaussian-Tail Extensions: the Gaussian Reference Measure, Marginals, Densities, Relative Entropy, Relative Score and the Noise Wasserstein Distance §heads, and $E_{n}(\tilde{\mu}_{n})$ is its Gaussian-tail extension at level $n$.

1. (Membership)¶ For every $n\in\mathbb{N}$, the measures $\mu$ and $E_{n}(\tilde{\mu}_{n})$ belong to $\mathcal{P}^{a}_{\rho}$, so that $W_{a}(\mu,E_{n}(\tilde{\mu}_{n}))$ is defined.

2. (The estimate)¶ For every $n\in\mathbb{N}$,

$$W_{a}\bigl(\mu,E_{n}(\tilde{\mu}_{n})\bigr)^{2}\le2\kappa\,\bigl(H(\mu\,|\,\gamma_{c})-H(\tilde{\mu}_{n}\,|\,\tilde{\gamma}_{n})\bigr).$$

3. (Convergence)¶ $W_{a}\bigl(\mu,E_{n}(\tilde{\mu}_{n})\bigr)\to0$ as $n\to\infty$.