Wasserstein Estimates for Affine Push-Forwards, First and Quadratic Moments, and the Cost of a Joint Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n,d\in\mathbb{N}$. By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric, $W_{2}$ is a metric on $\Sigma_{m}$; Cauchy sequences in $\Sigma_{m}$ are those of the metric space $(\Sigma_{m},W_{2})$ as in Cauchy Sequence in a Metric Space, and Cauchy sequences of real numbers are those of Cauchy Sequence of Real Numbers. Affine data $T$, their affine substitutions $\sigma_{T}$ and their norms $\lVert T\rVert$ are those of Affine Data and Affine Substitutions of Noncommutative Polynomials, and $\mathrm{m}_{i}$, $\mathrm{m}_{ij}$ are the first and quadratic moments. For $\lambda\in\Sigma_{m}$ the number $M(\lambda)$ is real and nonnegative by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, and $M(\lambda)^{1/2}$ is its nonnegative square root.

1. (Affine push-forwards)¶ For every affine datum $T$ from $m$ to $n$ variables and all $\mu,\nu\in\Sigma_{m}$,

2. (Moments)¶ For all $\mu,\nu\in\Sigma_{m}$ and $i,j\in[m]$,

3. (Cauchy sequences)¶ Let $(\lambda_{k})_{k\in\mathbb{N}}$ be a Cauchy sequence in $\Sigma_{m}$. Then $(\lambda_{k}\circ\sigma_{T})_{k\in\mathbb{N}}$ is a Cauchy sequence in $\Sigma_{n}$ for every affine datum $T$ from $m$ to $n$ variables, and for all $i,j\in[m]$ the real sequences $(\mathrm{m}_{i}(\lambda_{k}))_{k\in\mathbb{N}}$ and $(\mathrm{m}_{ij}(\lambda_{k}))_{k\in\mathbb{N}}$ are Cauchy sequences of real numbers.

4. (Cost)¶ Let $\gamma\in\Sigma_{2d}$, where $2d=d+d$, and let $\iota^{1},\iota^{2}$ and $I$ be the marginal substitutions and the cost. Then $\gamma\circ\iota^{1},\gamma\circ\iota^{2}\in\Sigma_{d}$, $\gamma\in\Pi(\gamma\circ\iota^{1},\gamma\circ\iota^{2})$, and $W_{2}(\gamma\circ\iota^{1},\gamma\circ\iota^{2})^{2}\le I(\gamma)$.