Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $m\in\mathbb{N}$ satisfy $1\le m$ and let $d_{E}$ denote the Euclidean distance, on $\mathbb{R}^{m}$ and on $\mathbb{R}^{m+m}$ as the dimension of its arguments requires; tightness is that of Tight Family of Borel Measures on a Metric Space (i) for the metric space named. Second moments $M_{2}$ and the sets $\mathcal{P}_{2}$ are those of The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment, and couplings and the sets $\Pi(\mu,\nu)$ are those of Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i).

Then the following hold.

1. (Bounded second moments give tightness)(i) Let $\mathcal{M}$ be a subset of $\mathcal{P}(\mathbb{R}^{m})$ and let $R\in\mathbb{R}$ satisfy $M_{2}(\mu)\le R$ for every $\mu\in\mathcal{M}$. Then $\mathcal{M}$ is tight in $(\mathbb{R}^{m},d_{E})$.

2. (Couplings)(ii) Let $\mu,\nu\in\mathcal{P}(\mathbb{R}^{m})$. Then every $\pi\in\Pi(\mu,\nu)$ satisfies

$$M_{2}(\pi)=M_{2}(\mu)+M_{2}(\nu)$$

in $[0,\infty]$. If moreover $\mu\in\mathcal{P}_{2}(\mathbb{R}^{m})$ and $\nu\in\mathcal{P}_{2}(\mathbb{R}^{m})$, then $\Pi(\mu,\nu)$ is tight in $(\mathbb{R}^{m+m},d_{E})$.