has also been reported in [24, 31, 43, 57, 80]. ∎

###  A.1. Support Functions of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ and
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$

We now show that the support functions of the convex sets
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ and
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be computed
efficiently. This will allow us to examine the Gelbrich risk of nonlinear
portfolio loss functions in Section 5 and the Gelbrich mean-variance risk in
Appendix E. The support functions of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ and
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ are also relevant for
robust optimization. For example, a robust constraint that requires a concave
function $h(\mu,\Sigma)$ to be nonpositive for all
$(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be
reformulated as a convex constraint that involves the convex conjugate of
$-h(\mu,\Sigma)$ as well as the support function of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ [4, Theorem 2]. Formally,
we have

$\displaystyle h(\mu,\Sigma)\leq 0\quad$
$\displaystyle\forall(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})\quad$
$\displaystyle\iff\quad\exists\;q\in\mathbb{R}^{n},Q\in\mathbb{S}^{n}\text{
such that
}(-h)^{*}(q,Q)+\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)\leq
0.$

This constraint is computationally tractable for many commonly used constraint
functions because the support function of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be represented as the
optimal value of a tractable conic minimization problem.

######  Proposition 13.

For any $\rho\geq 0$, $q\in\mathbb{R}^{n}$ and $Q\in\mathbb{S}^{n}$, the
support function of $\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$
satisfies

$\displaystyle\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\inf&q^{\top}\widehat{\mu}+\tau+\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\operatorname{Tr}\big{[}Z\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\tau\in\mathbb{R}_{+},\;Z\in\mathbb{S}^{n}_{+}\\\\[4.30554pt]
&\begin{bmatrix}\gamma I-Q&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\quad\Bigg{\|}\begin{bmatrix}q\\\
\tau-\gamma\end{bmatrix}\Bigg{\|}\leq\tau+\gamma.\end{array}\right.$

###### Proof of Proposition 13.

Evaluating the support function of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ at a given point
$(q,Q)\in\mathbb{R}^{n}\times\mathbb{S}^{n}$ amounts to solving the finite
convex program

$\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\sup\limits_{\mu,\Sigma\succeq
0}&q^{\top}\mu+\operatorname{Tr}\big{[}Q\Sigma\big{]}\\\
\operatorname{s.t.}&\|\mu-\widehat{\mu}\|^{2}+\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}\leq\rho^{2}.\end{array}\right.$

Using the semidefinite programming reformulation for the squared Gelbrich
distance on the right hand side of (11) and introducing an auxiliary variable
$M$ that equals $\mu\mu^{\top}$ at optimality, we then obtain

$\displaystyle\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=$
$\displaystyle\left\\{\begin{array}[]{cl}\sup\limits_{\mu,C,\Sigma\succeq
0}&q^{\top}\mu+\operatorname{Tr}\big{[}Q\Sigma\big{]}\\\
\operatorname{s.t.}&\operatorname{Tr}\big{[}\mu\mu^{\top}-2\widehat{\mu}\mu^{\top}+\widehat{\mu}\widehat{\mu}^{\top}\big{]}+\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2C\big{]}\leq\rho^{2}\\\\[4.30554pt]
&\begin{bmatrix}\Sigma&C\\\ C^{\top}&\widehat{\Sigma}\end{bmatrix}\succeq
0\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\sup&q^{\top}\mu+\operatorname{Tr}\big{[}Q\Sigma\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\mu\in\mathbb{R}^{n},\,C\in\mathbb{R}^{n\times
n},\,\Sigma\in\mathbb{S}_{+}^{n},\,M\in\mathbb{S}_{+}^{n}\\\\[4.30554pt]
&\operatorname{Tr}\big{[}M-2\widehat{\mu}\mu^{\top}\big{]}+\|\widehat{\mu}\|^{2}+\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2C\big{]}\leq\rho^{2}\\\\[4.30554pt]
&\begin{bmatrix}\Sigma&C\\\ C^{\top}&\widehat{\Sigma}\end{bmatrix}\succeq
0,~{}\begin{bmatrix}M&\mu\\\ \mu^{\top}&1\end{bmatrix}\succeq
0.\end{array}\right.$

By strong conic programming duality [5, Theorem 1.4.2], the resulting
semidefinite program is equivalent to

$\displaystyle\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=$
$\displaystyle\left\\{\begin{array}[]{cl}\inf&\gamma\left(\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\right)+\operatorname{Tr}\big{[}A_{22}\widehat{\Sigma}\big{]}+\beta\\\\[4.30554pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},~{}\begin{bmatrix}A_{11}&\gamma
I\\\ \gamma
I&A_{22}\end{bmatrix}\in\mathbb{S}_{+}^{2n},~{}\begin{bmatrix}B&\gamma\widehat{\mu}+\frac{q}{2}\\\
\left(\gamma\widehat{\mu}+\frac{q}{2}\right)^{\top}&\beta\end{bmatrix}\in\mathbb{S}_{+}^{n+1}\\\
&A_{11}\preceq\gamma I-Q,~{}B\preceq\gamma I.\end{array}\right.$ (15)

Strong duality holds because $\gamma=\max\\{\lambda_{\max}(Q),0\\}+2$,
$A_{11}=I$, $A_{22}=2\gamma^{2}I$, $B=I$ and
$\beta=\|\gamma\widehat{\mu}+\frac{q}{2}\|^{2}+1$ represents a Slater point
for the dual problem. Next, we simplify the semidefinite program (15) by
eliminating the decision variables $A_{11}$ and $B$, each of which appears in
two opposing matrix inequalities. If $\gamma>0$, then $A_{22}$ has full rank
thanks to [7, Corollary 8.2.2]. In this case, we obtain the following
equivalences by Schur complementing the two matrix inequalities involving
$A_{11}$.

$\displaystyle\begin{array}[]{ll}\begin{bmatrix}A_{11}&\gamma I\\\ \gamma
I&A_{22}\end{bmatrix}\succeq 0,~{}A_{11}\preceq\gamma
I-Q&\iff\gamma^{2}A_{22}^{-1}\preceq A_{11}\preceq\gamma I-Q\\\
&\iff\begin{bmatrix}\gamma I-Q&\gamma I\\\\[4.30554pt] \gamma
I&A_{22}\end{bmatrix}\succeq 0\end{array}$ (18)

If $\gamma=0$, on the other hand, we trivially have

$\displaystyle\begin{bmatrix}A_{11}&\gamma I\\\ \gamma
I&A_{22}\end{bmatrix}\succeq 0,~{}A_{11}\preceq\gamma I-Q\iff 0\preceq
A_{11}\preceq-Q,~{}A_{22}\succeq 0\iff\begin{bmatrix}-Q&0\\\\[4.30554pt]
0&A_{22}\end{bmatrix}\succeq 0.$

Therefore, all equivalences in (18) hold for any $\gamma\geq 0$. If $\beta>0$,
then Schur complementing the two matrix inequalities involving $B$ yields

$\displaystyle\begin{bmatrix}B&\gamma\widehat{\mu}+\frac{q}{2}\\\
\left(\gamma\widehat{\mu}+\frac{q}{2}\right)^{\top}&\beta\end{bmatrix}\succeq
0,~{}B\preceq\gamma I$
$\displaystyle\iff\frac{1}{\beta}\left(\gamma\widehat{\mu}+\frac{q}{2}\right)\left(\gamma\widehat{\mu}+\frac{q}{2}\right)^{\top}\preceq
B\preceq\gamma I$ $\displaystyle\iff\begin{bmatrix}\gamma
I&\gamma\widehat{\mu}+\frac{q}{2}\\\\[4.30554pt]
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&\beta\end{bmatrix}\succeq 0.$ (19)

If $\beta=0$, on the other hand, we have

$\displaystyle\begin{bmatrix}B&\gamma\widehat{\mu}+\frac{q}{2}\\\
\left(\gamma\widehat{\mu}+\frac{q}{2}\right)^{\top}&\beta\end{bmatrix}\succeq
0,~{}B\preceq\gamma I$ $\displaystyle\iff 0\preceq B\preceq\gamma
I\iff\begin{bmatrix}\gamma I&0\\\\[4.30554pt] 0&0\end{bmatrix}\succeq 0,$

where the first equivalence holds because $\beta=0$ implies via the first
matrix inequality that $\gamma\widehat{\mu}+\frac{q}{2}=0$; see [7, Corollary
8.2.2]. Therefore, all equivalences in (19) hold for any $\beta\geq 0$. In
summary, we have thus shown that

$\displaystyle\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\inf\limits_{\gamma,\beta,A_{22}}&\gamma\left(\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\right)+\operatorname{Tr}\big{[}A_{22}\widehat{\Sigma}\big{]}+\beta\\\\[8.61108pt]
\operatorname{s.t.}&\begin{bmatrix}\gamma I-Q&\gamma I\\\\[4.30554pt] \gamma
I&A_{22}\end{bmatrix}\succeq 0,~{}\begin{bmatrix}\gamma
I&\gamma\widehat{\mu}+\frac{q}{2}\\\\[4.30554pt]
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&\beta\end{bmatrix}\succeq
0.\end{array}\right.$ (22)

For any fixed $\gamma\geq 0$, problem (22) decomposes into two separate
minimization problems over $\beta$ and $A_{22}$, respectively. If $\gamma>0$,
the partial minimization problem over $\beta$ reduces to

$\displaystyle\phantom{=}\inf\left\\{\beta:\begin{bmatrix}\gamma
I&\gamma\widehat{\mu}+\frac{q}{2}\\\\[4.30554pt]
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&\beta\end{bmatrix}\succeq 0\right\\}$
$\displaystyle=\inf\left\\{\beta:\beta-\gamma\|\widehat{\mu}\|^{2}-q^{\top}\widehat{\mu}-\frac{1}{4\gamma}\|q\|^{2}\geq
0\right\\}$
$\displaystyle=\inf\left\\{\eta+\gamma\|\widehat{\mu}\|^{2}:\eta\geq
q^{\top}\widehat{\mu}+\tau,\;\Bigg{\|}\begin{bmatrix}q\\\
\tau-\gamma\end{bmatrix}\Bigg{\|}\leq\tau+\gamma,\;\tau\geq 0\right\\},$

where the first equality exploits a standard Schur complement argument, and
the second equality follows from the substitution
$\eta\leftarrow\beta-\gamma\|\widehat{\mu}\|^{2}$ and from introducing an
auxiliary variable $\tau\geq 0$ subject to the hyperbolic constraint
$\tau\geq\|q\|^{2}/(4\gamma)$. Note also that the first and the last
minimization problems in the above expression remain equivalent when
$\gamma=0$. Similarly, if $\gamma>0$, then the partial minimization problem
over $A_{22}$ reduces to

$\displaystyle\phantom{=}\inf\left\\{\operatorname{Tr}\big{[}A_{22}\widehat{\Sigma}\big{]}:\begin{bmatrix}\gamma
I-Q&\gamma I\\\\[4.30554pt] \gamma I&A_{22}\end{bmatrix}\succeq 0\right\\}$
$\displaystyle=\inf\left\\{\operatorname{Tr}\big{[}Z\big{]}:Z\succeq\widehat{\Sigma}^{\frac{1}{2}}A_{22}\widehat{\Sigma}^{\frac{1}{2}},A_{22}\succeq\gamma^{2}(\gamma
I-Q)^{-1}\right\\}$
$\displaystyle=\inf\left\\{\operatorname{Tr}\big{[}Z\big{]}:Z\succeq\gamma\widehat{\Sigma}^{\frac{1}{2}}(\gamma
I-Q)^{-1}\gamma\widehat{\Sigma}^{\frac{1}{2}}\right\\}$
$\displaystyle=\inf\left\\{\operatorname{Tr}\big{[}Z\big{]}:\begin{bmatrix}\gamma
I-Q&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\\[4.30554pt]
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq 0\right\\},$

where the first and the third equalities follow from Schur complement
arguments. As $\widehat{\Sigma}\succ 0$ by assumption, the first and the last
minimization problems in the above expression remain equivalent when
$\gamma=0$. The claim then follows by substituting the reformulated partial
minimization problems into (22) and eliminating $\eta$. ∎

The next lemma shows that the unique maximizer of the optimization problem
defining the support function of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be computed in quasi-
closed form.

######  Lemma 2.

Suppose that $\widehat{\Sigma}\succ 0$, $\rho>0$ and either $q\neq 0$ or
$\lambda_{\max}(Q)>0$. Then the optimization problem

$\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}~{}q^{\top}\mu+\operatorname{Tr}\big{[}Q\Sigma\big{]}$
(23a) is uniquely solved by
$\displaystyle\mu^{\star}=\widehat{\mu}+\frac{q}{2\gamma^{\star}}\quad\text{and}\quad\Sigma^{\star}=\left(I-\frac{Q}{\gamma^{\star}}\right)^{-1}\widehat{\Sigma}\left(I-\frac{Q}{\gamma^{\star}}\right)^{-1},$
(23b) where $\gamma^{\star}>\max\\{\lambda_{\max}(Q),0\\}$ is the unique
solution of the nonlinear algebraic equation
$\displaystyle\frac{\|q\|^{2}}{4\gamma^{2}}+\operatorname{Tr}\big{[}\widehat{\Sigma}\left(I-\gamma(\gamma
I-Q)^{-1}\right)^{2}\big{]}=\rho^{2}.$ (23c)

In addition, if $Q\succeq 0$ then we have
$\Sigma^{\star}\succeq\lambda_{\min}(\widehat{\Sigma})I$.

###### Proof of Lemma 2.

From the proof of Proposition 13 we know that problem (23a) is equivalent to
the semidefinite program (22). We first prove that $\gamma>0$ for any
$(\gamma,\beta,A_{22})$ feasible in (22). To see this, assume for the sake of
argument that $\gamma=0$, in which case the two matrix inequalities in (22)
reduce to

$\displaystyle\begin{bmatrix}-Q&0\\\\[4.30554pt] 0&A_{22}\end{bmatrix}\succeq
0\quad\text{and}\quad\begin{bmatrix}0&{q}/{2}\\\\[4.30554pt]
\left({q}/{2}\right)^{\top}&\beta\end{bmatrix}\succeq 0.$

However, these constraints are not satisfiable by any $\beta$ and $A_{22}$
under our assumption that either $q\neq 0$ or $\lambda_{\max}(Q)>0$.
Similarly, one can show that $\gamma>\lambda_{\max}(Q)$ for any
$(\gamma,\beta,A_{22})$ feasible in (22). Indeed, we have

$\displaystyle\begin{bmatrix}\gamma I-Q&\gamma I\\\ \gamma
I&A_{22}\end{bmatrix}\succeq 0,~{}\gamma>0\quad\implies\quad\gamma I-Q\succ
0\quad\implies\quad\gamma>\lambda_{\max}(Q),$

where the first implication follows from [7, Corollary 8.2.2], which requires
$\gamma I-Q$ to have full rank whenever the off-diagonal block $\gamma I$ has
full rank. Schur complementing the two matrix inequalities in (22) yields

$A_{22}\succeq\gamma^{2}(\gamma
I-Q)^{-1}\quad\text{and}\quad\textstyle\beta\geq\|\gamma\widehat{\mu}+\frac{q}{2}\|^{2}/\gamma,$

which is possible because $\gamma>\max\\{\lambda_{\max}(Q),0\\}$. Using these
relations to eliminate $\beta$ and $A_{22}$ from (22) yields

$\displaystyle\begin{array}[]{ll}&\phantom{=}\displaystyle\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)\\\
&=\displaystyle\inf\limits_{\begin{subarray}{c}\gamma>0\gamma>\lambda_{\max}(Q)\end{subarray}}q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{4\gamma}+\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\gamma^{2}\operatorname{Tr}\big{[}(\gamma
I-Q)^{-1}\widehat{\Sigma}\big{]}.\end{array}$ (26)

In the following we use $f(\gamma)$ to denote the objective function of
problem (26). As $\widehat{\Sigma}\succ 0$, $f(\gamma)$ satisfies

$f(\gamma)\geq
q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{4\gamma}+\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\lambda_{\min}(\widehat{\Sigma})\gamma^{2}\operatorname{Tr}\big{[}(\gamma
I-Q)^{-1}\big{]}$

for all $\gamma\geq\max\\{\lambda_{\max}(Q),0\\}$. Thus, $f(\gamma)$ diverges
as $\gamma$ decreases to $\max\\{\lambda_{\max}(Q),0\\}$. Similarly, since
$\rho>0$, one readily verifies that $f(\gamma)$ grows indefinitely as $\gamma$
increases to infinity. Noting that $f(\gamma)$ is smooth and strictly convex
on its domain, these insights reveal that problem (26) has a unique minimizer
$\gamma^{\star}\in(\max\\{0,\lambda_{\max}(Q)\\},\infty)$. This minimizer can
be found by solving the problem’s first-order optimality condition
$f^{\prime}(\gamma^{\star})=0$, where

$\displaystyle f^{\prime}(\gamma)$
$\displaystyle=-\frac{\|q\|^{2}}{4\gamma^{2}}+\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+2\gamma\operatorname{Tr}\big{[}(\gamma
I-Q)^{-1}\widehat{\Sigma}\big{]}-\gamma^{2}\operatorname{Tr}\big{[}(\gamma
I-Q)^{-2}\widehat{\Sigma}\big{]}$
$\displaystyle=\rho^{2}-\frac{\|q\|^{2}}{4\gamma^{2}}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{(}I-\gamma(\gamma
I-Q)^{-1}\big{)}^{2}\big{]}$

denotes the derivative of $f(\gamma)$. Thus, the algebraic equation (23c)
represents the (necessary and sufficient) first-order optimality condition of
problem (26). In the remainder of the proof, we demonstrate that
$(\mu^{\star},\Sigma^{\star})$ as defined in (23b) constitutes a global
maximizer of problem (23a). To see this, note first that
$\Sigma^{\star}\succeq 0$ and

$\displaystyle\mathds{G}^{2}\big{(}(\mu^{\star},\Sigma^{\star}),(\widehat{\mu},\widehat{\Sigma})\big{)}^{2}=\frac{\|q\|^{2}}{4(\gamma^{\star})^{2}}+\operatorname{Tr}\big{[}\widehat{\Sigma}(I-\gamma^{\star}(\gamma^{\star}I-Q)^{-1})^{2}\big{]}=\rho^{2},$

where the first equality follows from (23b) and the definition of the Gelbrich
distance, whereas the second equality holds because $\gamma^{\star}$ solves
(23c). Thus, $(\mu^{\star},\Sigma^{\star})$ is feasible in problem (23a).
Furthermore, the objective function value of $(\mu^{\star},\Sigma^{\star})$ in
(23a) evaluates to

$\displaystyle\phantom{=}q^{\top}\mu^{\star}+\operatorname{Tr}\big{[}Q\Sigma^{\star}\big{]}$
$\displaystyle=q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{2\gamma^{\star}}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}Q(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
$\displaystyle=q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{2\gamma^{\star}}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(Q-\gamma^{\star}I+\gamma^{\star}I)(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
$\displaystyle=q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{2\gamma^{\star}}-(\gamma^{\star})^{2}\operatorname{Tr}\big{[}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}+(\gamma^{\star})^{3}\operatorname{Tr}\big{[}\widehat{\Sigma}(\gamma
I-Q)^{-2}\big{]}$
$\displaystyle=q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{2\gamma^{\star}}+\gamma^{\star}\left(\rho^{2}-\frac{\|q\|^{2}}{4(\gamma^{\star})^{2}}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\right)+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}$
$\displaystyle=q^{\top}\widehat{\mu}+\frac{\|q\|^{2}}{4\gamma^{\star}}+\gamma^{\star}\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}=\delta^{*}_{\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q),$

where the fourth equality holds because $\gamma^{\star}>0$ solves (23c), and
the last equality follows from the optimality of $\gamma^{\star}$ in (26).
Thus, $(\mu^{\star},\Sigma^{\star})$ is optimal in (23a). As problem (23a) has
a linear objective function and a strictly convex feasible set, the maximizwer
$(\mu^{\star},\Sigma^{\star})$ is unique. To complete the proof, note that if
$Q\succeq 0$, then $(I-Q/\gamma^{\star})^{-1}\succeq I$ because
$\gamma^{\star}I\succ Q$, and thus it is easy to verify that
$\Sigma^{\star}\succeq\lambda_{\min}(\widehat{\Sigma})I$. ∎

Next, we derive a tractable reformulation for the support function of
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$.

######  Proposition 14.

For any $\rho\in\mathbb{R}_{+}$, $q\in\mathbb{R}^{n}$ and
$Q\in\mathbb{S}^{n}$, the support function of
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ satisfies

$\displaystyle\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\inf&\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\operatorname{Tr}\big{[}Z\big{]}+z\\\\[8.61108pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\begin{bmatrix}\gamma
I-Q&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\\[4.30554pt]
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Q&\gamma\widehat{\mu}+\frac{q}{2}\\\\[4.30554pt]
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&z\end{bmatrix}\succeq
0.\end{array}\right.$

###### Proof of Proposition 14.

Evaluating the support function of
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ at
$(q,Q)\in\mathbb{R}^{n}\times\mathbb{S}^{n}$ amounts to solving a maximization
problem

$\displaystyle\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\sup&\mu^{\top}q+\operatorname{Tr}\big{[}MQ\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\mu\in\mathbb{R}^{n},\;M\in\mathbb{S}_{+}^{n},\;\mathds{G}^{2}\big{(}(\mu,M-\mu\mu^{\top}),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\rho^{2}.\end{array}\right.$

The semidefinite programming representation of the squared Gelbrich distance
in (11) then yields

$\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\left\\{\begin{array}[]{cl}\sup&\mu^{\top}q+\operatorname{Tr}\big{[}MQ\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\mu\in\mathbb{R}^{n},\;M\in\mathbb{S}_{+}^{n},\;C\in\mathbb{R}^{n\times
n}\\\\[4.30554pt]
&\|\mu-\widehat{\mu}\|^{2}+\operatorname{Tr}\big{[}(M-\mu\mu^{\top})+\widehat{\Sigma}-2C\big{]}\leq\rho^{2}\\\\[4.30554pt]
&\begin{bmatrix}M-\mu\mu^{\top}&C\\\
C^{\top}&\widehat{\Sigma}\end{bmatrix}\succeq 0.\end{array}\right.$

Note that the first constraint is equivalent to
$\operatorname{Tr}\big{[}M-2\mu\widehat{\mu}^{\top}-2C\big{]}\leq\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}$.
Introducing a new decision variable $U\in\mathbb{S}_{+}^{n}$ with
$U\succeq\mu\mu^{\top}$ and then applying a Schur complement argument, the
above optimization problem can be recast as the following semidefinite
program.

$\begin{array}[]{cl}\sup&\mu^{\top}q+\operatorname{Tr}\big{[}MQ\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\mu\in\mathbb{R}^{n},\;M\in\mathbb{S}_{+}^{n},\;U\in\mathbb{S}_{+}^{n},\;C\in\mathbb{R}^{n\times
n}\\\\[4.30554pt]
&\operatorname{Tr}\big{[}M-2\mu\widehat{\mu}^{\top}-2C\big{]}\leq\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\\\\[4.30554pt]
&\begin{bmatrix}M-U&C\\\ C^{\top}&\widehat{\Sigma}\end{bmatrix}\succeq
0,\;\begin{bmatrix}U&\mu\\\ \mu^{\top}&1\end{bmatrix}\succeq 0\end{array}$

By strong conic duality [5, Theorem 1.4.2], the above semidefinite program
admits the dual

$\displaystyle\begin{array}[]{cl}\displaystyle\inf&\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\operatorname{Tr}\big{[}\widehat{\Sigma}A_{22}\big{]}+\beta\\\\[8.61108pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\begin{bmatrix}A_{11}&\gamma
I\\\ \gamma I&A_{22}\end{bmatrix}\succeq
0,\;\begin{bmatrix}B&\gamma\widehat{\mu}+\frac{q}{2}\\\
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&\beta\end{bmatrix}\succeq 0,\;\gamma
I-Q\succeq A_{11}\succeq B.\end{array}$

Strong duality holds because
$\gamma=\max\\{\lambda_{\max}(Q),0\\}+4,A_{11}=2I,A_{22}=\gamma^{2}I,B=I,\beta=\|2\gamma\widehat{\mu}+q\|^{2}$
represents a Slater point for the dual problem. As in the proof of Proposition
13, one can eliminate the decision variables $A_{11}$ and $B$ by showing that
the matrix inequalities $\gamma I-Q\succeq A_{11}\succeq B$ are binding at
optimality. Thus, we can further simplify the dual problem to

$\displaystyle\begin{array}[]{cl}\displaystyle\inf&\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\operatorname{Tr}\big{[}\widehat{\Sigma}A_{22}\big{]}+\beta\\\\[8.61108pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\begin{bmatrix}\gamma I-Q&\gamma
I\\\ \gamma
I&A_{22}\end{bmatrix}\in\mathbb{S}_{+}^{2n},\;\begin{bmatrix}\gamma
I-Q&\gamma\widehat{\mu}+\frac{q}{2}\\\
(\gamma\widehat{\mu}+\frac{q}{2})^{\top}&\beta\end{bmatrix}\in\mathbb{S}_{+}^{n+1}.\end{array}$

The claim then follows by renaming
$\widehat{\Sigma}^{\frac{1}{2}}A_{22}\widehat{\Sigma}^{\frac{1}{2}}$ and
$\beta$ as $Z$ and $z$, respectively. ∎

In analogy to Lemma 2, the next lemma shows that the unique maximizer of the
optimization problem defining the support function of
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be computed in quasi-
closed form.

######  Lemma 3.

Suppose that $\widehat{\Sigma}\succ 0$, $\rho>0$ and $\lambda_{\max}(Q)>0$.
Then the optimization problem

$\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)=\sup\limits_{(\mu,M)\in\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}~{}q^{\top}\mu+\operatorname{Tr}\big{[}QM\big{]}$
(27a) is uniquely solved by
$\displaystyle\mu^{\star}=(\gamma^{\star}I-Q)^{-1}\left(\gamma^{\star}\widehat{\mu}+\frac{q}{2}\right)\quad\text{and}\quad
M^{\star}=\left(I-\frac{Q}{\gamma^{\star}}\right)^{-1}\widehat{\Sigma}\left(I-\frac{Q}{\gamma^{\star}}\right)^{-1}+\mu^{\star}{\mu^{\star}}^{\top},$
(27b) where $\gamma^{\star}>\lambda_{\max}(Q)$ is the unique solution of the
nonlinear algebraic equation $\displaystyle\|\widehat{\mu}-(\gamma
I-Q)^{-1}(q/2+\gamma\widehat{\mu})\|^{2}+\operatorname{Tr}\big{[}\widehat{\Sigma}\left(I-\gamma(\gamma
I-Q)^{-1}\right)^{2}\big{]}=\rho^{2}.$ (27c)

###### Proof of Lemma 3.

Using Schur complement argument, one can further simplifies the result of
Proposition 14 and conclude that the support function of
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is equivalent to the
optimal value of the following univariate optimization problem

$\begin{array}[]{cl}\inf&\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\gamma^{2}\operatorname{Tr}\big{[}(\gamma
I-Q)^{-1}\widehat{\Sigma}\big{]}+(\gamma\widehat{\mu}+\frac{q}{2})^{\top}[\gamma
I-Q]^{-1}(\gamma\widehat{\mu}+\frac{q}{2})\\\
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\gamma I\succ Q.\end{array}$
(28)

Denote momentarily the objective function of (28) as $f(\gamma)$. The gradient
of $f$ for any feasible solution $\gamma$ satisfies

$\displaystyle\nabla_{\gamma}f$
$\displaystyle=\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\left(I-\gamma(\gamma
I-Q)^{-1}\right)^{2}\big{]}-\Big{\|}\widehat{\mu}-(\gamma
I-Q)^{-1}\left(\gamma\widehat{\mu}+\frac{q}{2}\right)\Big{\|}^{2}.$

Thus, if $\gamma^{\star}$ solves the nonlinear algebraic equation (27c), then
$\gamma^{\star}$ also solves the first-order optimality condition of problem
(28). This in turn implies that $\gamma^{\star}$ is the minimizer of problem
(28).

As we have assumed that $\lambda_{\max}(Q)>0$, it is easy to verify that as
$\gamma$ approaches $\lambda_{\max}(Q)$, $\nabla_{\gamma}f$ tends to
$-\infty$, and as $\gamma$ tends to infinity, $\nabla_{\gamma}f$ tends to
$\rho^{2}>0$. This asserts that there exists a finite value $\gamma^{\star}$
that solves (27c). Let
$\Sigma^{\star}=M^{\star}-\mu^{\star}{\mu^{\star}}^{\top}$. Notice that

$\mathds{G}\big{(}(\mu^{\star},\Sigma^{\star}),(\widehat{\mu},\widehat{\Sigma})\big{)}^{2}=\Big{\|}(\gamma^{\star}I-Q)^{-1}\left(\gamma^{\star}\widehat{\mu}+\frac{q}{2}\right)-\widehat{\mu}\Big{\|}^{2}+\operatorname{Tr}\big{[}\widehat{\Sigma}\left(I-\gamma^{\star}(\gamma^{\star}I-Q)^{-1}\right)^{2}\big{]}=\rho^{2},$

where the first equality follows from substituting the value of $\mu^{\star}$
and $\Sigma^{\star}$ from (27b) into the definition of the Gelbrich distance
$\mathds{G}$, and the second equality is because $\gamma^{\star}$ solves
(27c). This implies that
$(\mu^{\star},\Sigma^{\star})\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$.
Next, we show that $(\mu^{\star},M^{\star})$ will attain the value
$\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(q,Q)$.
Observe that

$\displaystyle\operatorname{Tr}\big{[}Q\Sigma^{\star}\big{]}=$
$\displaystyle\operatorname{Tr}\big{[}Q(\gamma^{\star})^{2}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
(29a) $\displaystyle=$
$\displaystyle\operatorname{Tr}\big{[}(Q-\gamma^{\star}I+\gamma^{\star}I)(\gamma^{\star})^{2}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
$\displaystyle=$
$\displaystyle-(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}+(\gamma^{\star})^{3}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
$\displaystyle=$
$\displaystyle\gamma^{\star}\big{(}\rho^{2}-\|\widehat{\mu}-(\gamma^{\star}I-Q)^{-1}(\gamma^{\star}\widehat{\mu}+q/2)\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+\gamma^{\star}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}\big{)}$
(29b) $\displaystyle=$
$\displaystyle\gamma^{\star}\big{(}\rho^{2}-\|\widehat{\mu}-\mu^{\star}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+\gamma^{\star}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}\big{)}$
(29c) $\displaystyle=$
$\displaystyle\gamma^{\star}\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}+2\gamma^{\star}\widehat{\mu}^{\top}\mu^{\star}-\gamma^{\star}\|\mu^{\star}\|^{2},$

where equality (29a) is from the definition of $\Sigma^{\star}$, equality
(29b) follows from the fact that $\gamma^{\star}$ solves (27c) and thus we can
write

$\displaystyle-\gamma^{\star}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}(\gamma^{\star}I-Q)^{-1}\big{]}$
$\displaystyle\hskip
56.9055pt=\rho^{2}-\|\widehat{\mu}-(\gamma^{\star}I-Q)^{-1}(q/2+\gamma^{\star}\widehat{\mu})\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+\gamma^{\star}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}.$

Finally, equality (29c) is from the definition of $\mu^{\star}$. We thus find

$\displaystyle
q^{\top}\mu^{\star}+\operatorname{Tr}\big{[}Q(\Sigma^{\star}+\mu^{\star}(\mu^{\star})^{\top})\big{]}$
$\displaystyle=\gamma^{\star}\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}+2(\gamma^{\star}\widehat{\mu}+q/2)^{\top}\mu^{\star}+(\mu^{\star})^{\top}(Q-\gamma^{\star}I)\mu^{\star}$
$\displaystyle=\gamma^{\star}\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+(\gamma^{\star})^{2}\operatorname{Tr}\big{[}(\gamma^{\star}I-Q)^{-1}\widehat{\Sigma}\big{]}+(\gamma^{\star}\widehat{\mu}+q/2)^{\top}(\gamma^{\star}I-Q)^{-1}(\gamma^{\star}\widehat{\mu}+q/2)$

which equals the optimal value of the dual program (28) because
$\gamma^{\star}$ is also the minimizer of problem (28). This concludes the
proof. ∎

##  Appendix B Proofs of Section 3

We give a short proof of Theorem 1 in order to keep this paper self-contained.

###### Proof of Theorem 1.

By the definition of the 2-Wasserstein distance, we have

$\displaystyle\mathds{W}^{2}(\mathbb{Q}_{1},\mathbb{Q}_{2})=$
$\displaystyle\displaystyle\min_{\pi\in\Pi(\mathbb{Q}_{1},\mathbb{Q}_{2})}~{}\displaystyle\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\|\xi_{1}-\xi_{2}\|^{2}\pi(\mathrm{d}\xi_{1},\mathrm{d}\xi_{2})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\displaystyle\min&\displaystyle\|\mu_{1}-\mu_{2}\|^{2}+\operatorname{Tr}\big{[}\Sigma_{1}+\Sigma_{2}-2C\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\pi\in\Pi(\mathbb{Q}_{1},\mathbb{Q}_{2}),\;C\in\mathbb{R}^{n\times
n}\\\\[4.30554pt]
&\displaystyle\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\begin{bmatrix}\xi_{1}\\\
\xi_{2}\end{bmatrix}\begin{bmatrix}\xi_{1}\\\
\xi_{2}\end{bmatrix}^{\top}\pi(\mathrm{d}\xi_{1},\mathrm{d}\xi_{2})=\begin{bmatrix}\Sigma_{1}&C\\\
C^{\top}&\Sigma_{2}\end{bmatrix},\quad\begin{bmatrix}\Sigma_{1}&C\\\
C^{\top}&\Sigma_{2}\end{bmatrix}\succeq 0,\end{array}\right.$

where the second equality follows from the observations that $C$ is uniquely
determined by $\pi$ (thanks to the equality constraint in the last line) and
that the conic constraint is redundant (because the second-order momement
matrix of $\pi$ is always positive semidefinite). Relaxing the last
optimization problem by removing all constraints that involve the original
decision variable $\pi$, which in turn allows us to remove $\pi$ itself, we
find

$\mathds{W}^{2}(\mathbb{Q}_{1},\mathbb{Q}_{2})\geq\left\\{\begin{array}[]{cl}\displaystyle\min_{C\in\mathbb{R}^{n\times
n}}&\displaystyle\|\mu_{1}-\mu_{2}\|^{2}+\operatorname{Tr}\big{[}\Sigma_{1}+\Sigma_{2}-2C\big{]}\\\
\operatorname{s.t.}&\begin{bmatrix}\Sigma_{1}&C\\\
C^{\top}&\Sigma_{2}\end{bmatrix}\succeq 0.\end{array}\right.$

By [68, Proposition 2], the above semidefinite program can be solved
analytically, and its optimal value is given by
$\mathds{G}^{2}((\mu_{1},\Sigma_{1}),(\mu_{2},\Sigma_{2}))$. We emphasize that
the same analytical formula has also been reported in [24, 31, 43, 57, 80].
The claim then follows by taking square roots on both sides of (11). ∎

Our proof of Theorem 2 relies on the following preparatory lemma.

######  Lemma 4 (2-Wasserstein distances of perfectly correlated distributions
[25, Theorem 2.13]).

If $\mathbb{Q}_{1}\in\mathcal{M}_{2}$ has mean 0 and covariance matrix
$\Sigma_{1}\in\mathbb{S}_{+}^{n}$, and if $\mathbb{Q}_{2}=\mathbb{Q}_{1}\circ
g^{-1}$ is the pushforward distribution of $\mathbb{Q}_{1}$ under the positive
semidefinite linear transformation $g(\xi)=A\xi$ with
$A\in\mathbb{S}_{+}^{n}$, then

$\mathds{W}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\sqrt{\mathds{E}_{\mathbb{Q}_{1}}\left[\|\xi-A\xi\|^{2}\right]}=\sqrt{\operatorname{Tr}\big{[}\Sigma_{1}+A\Sigma_{1}A-2\Sigma_{1}^{\frac{1}{2}}A\Sigma_{1}^{\frac{1}{2}}\big{]}}.$
(30)

Lemma 4 implies that if $\xi$ follows a distribution $\mathbb{Q}_{1}$ with
vanishing mean and if $\mathbb{Q}_{2}$ is defined as the distribution of
$A\xi$ for some $A\in\mathbb{S}_{+}^{n}$, then the 2-Wasserstein distance
between the perfectly correlated distributions $\mathbb{Q}_{1}$ and
$\mathbb{Q}_{2}$ is given by (30). We can now use this (nontrivial) result to
prove Theorem 2.

###### Proof of Theorem 2.

By assumption, we have $\mathbb{Q}_{2}=\mathbb{Q}_{1}\circ f^{-1}$, where
$f(\xi)=A\xi+b$ is an affine transformation with parameters
$A\in\mathbb{S}_{+}^{n}$ and $b\in\mathbb{R}^{n}$. In the following, it will
be more convenient to re-express this affine transformation as
$f(\xi)=A(\xi-\mu_{1})+b^{\prime}$, which involves the auxiliary parameter
$b^{\prime}=A\mu_{1}+b$. Thus, we have

$\mu_{2}=\mathds{E}_{\mathbb{Q}_{2}}[\xi]=\mathds{E}_{\mathbb{Q}_{1}}[f(\xi)]=b^{\prime},$

where the three equalities follow from the definition of $\mu_{2}$, the
integration formula for pushforward distributions and the definitions of
$\mu_{1}$ and $f$, respectively. Similarly, we find

$\Sigma_{2}=\mathds{E}_{\mathbb{Q}_{2}}[(\xi-\mu_{2})(\xi-\mu_{2})^{\top}]=\mathds{E}_{\mathbb{Q}_{1}}[(f(\xi)-b^{\prime})(f(\xi)-b^{\prime})^{\top}]=A\Sigma_{1}A,$

where the second equality exploits our earlier insight that
$\mu_{2}=b^{\prime}$. Multiplying the above expression from both sides with
$\Sigma_{1}^{\frac{1}{2}}$ yields the quadratic equation
$\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}=(\Sigma_{1}^{\frac{1}{2}}A\Sigma_{1}^{\frac{1}{2}})^{2}$.
As $\Sigma_{1}\succ 0$ by assumption, this equation is uniquely solved by
$A=\Sigma_{1}^{-\frac{1}{2}}(\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}})^{\frac{1}{2}}\Sigma_{1}^{-\frac{1}{2}}$.
This confirms that the affine function $f$ is uniquely determined by the
first- and second-order moments of $\mathbb{Q}_{1}$ and $\mathbb{Q}_{2}$ and
that its parameters are given by (3).

Next, we define two distributions
$\overline{\mathbb{Q}}_{1},\overline{\mathbb{Q}}_{2}\in\mathcal{M}_{2}$
through the relations $\overline{\mathbb{Q}}_{1}[\xi\in
B]=\mathbb{Q}_{1}[(\xi+\mu_{1})\in B]$ and $\overline{\mathbb{Q}}_{2}[\xi\in
B]=\mathbb{Q}_{2}[(\xi+\mu_{2})\in B]$ for all Borel sets
$B\in\mathcal{B}(\mathbb{R}^{n})$. Thus, $\overline{\mathbb{Q}}_{1}$ and
$\overline{\mathbb{Q}}_{2}$ are obtained by shifting $\mathbb{Q}_{1}$ and
$\mathbb{Q}_{2}$ so that their mean vectors vanish. By construction, we then
have $\overline{\mathbb{Q}}_{2}=\overline{\mathbb{Q}}_{1}\circ g^{-1}$ where
$g(\xi)=A\xi$. By the definition of the 2-Wasserstein distance and the shifted
distributions $\overline{\mathbb{Q}}_{1}$ and $\overline{\mathbb{Q}}_{2}$, we
further have

$\mathds{W}^{2}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\|\mu_{1}-\mu_{2}\|^{2}+\mathds{W}^{2}(\overline{\mathbb{Q}}_{1},\overline{\mathbb{Q}}_{2}).$
(31)

Finally, as $\overline{\mathbb{Q}}_{2}=\overline{\mathbb{Q}}_{1}\circ g^{-1}$,
we may use Lemma 4 to conclude that

$\displaystyle\mathds{W}^{2}(\overline{\mathbb{Q}}_{1},\overline{\mathbb{Q}}_{2})$
$\displaystyle={\operatorname{Tr}\big{[}\Sigma_{1}+A\Sigma_{1}A-2\Sigma_{1}^{\frac{1}{2}}A\Sigma_{1}^{\frac{1}{2}}\big{]}}$
$\displaystyle=\operatorname{Tr}\big{[}\Sigma_{1}+\Big{(}\Sigma_{1}^{-\frac{1}{2}}\big{(}\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\Sigma_{1}^{-\frac{1}{2}}\Big{)}\Sigma_{1}\Big{(}\Sigma_{1}^{-\frac{1}{2}}\big{(}\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\Sigma_{1}^{-\frac{1}{2}}\Big{)}\big{]}$
$\displaystyle\quad-2\operatorname{Tr}\big{[}\Sigma_{1}^{\frac{1}{2}}\Big{(}\Sigma_{1}^{-\frac{1}{2}}\big{(}\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\Sigma_{1}^{-\frac{1}{2}}\Big{)}\Sigma_{1}^{\frac{1}{2}}\big{]}$
$\displaystyle=\operatorname{Tr}\big{[}\Sigma_{1}+\Sigma_{2}-2\big{(}\Sigma_{1}^{\frac{1}{2}}\Sigma_{2}\Sigma_{1}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]},$

where the second equality uses the expression for $A$ in (3). The claim then
follows by substituting the above expression into (31) and taking square roots
on both sides. ∎

###### Proof of Theorem 4.

By [11, Theorem 1], the Gelbrich distance satisfies

$\displaystyle\mathds{G}\big{(}(\widehat{\mu}_{N},\widehat{\Sigma}_{N}),(\mu,\Sigma)\big{)}$
$\displaystyle\leq\sqrt{\|\widehat{\mu}_{N}-\mu\|^{2}+\|\widehat{\Sigma}_{N}^{\frac{1}{2}}-\Sigma^{\frac{1}{2}}\|_{F}^{2}}$
$\displaystyle\leq\|\widehat{\mu}_{N}-\mu\|+\|\widehat{\Sigma}_{N}^{\frac{1}{2}}-\Sigma^{\frac{1}{2}}\|_{F}$
$\displaystyle\leq\|\widehat{\mu}_{N}-\mu\|+\frac{1}{\lambda_{\min}(\widehat{\Sigma}_{N})+\lambda_{\min}(\Sigma)}\,\|\widehat{\Sigma}_{N}-\Sigma\|_{F}$
$\displaystyle\leq\|\widehat{\mu}_{N}-\mu\|+\frac{1}{\lambda_{\min}(\Sigma)}\,\|\widehat{M}_{N}-M+\mu\mu^{\top}-\widehat{\mu}_{N}\widehat{\mu}_{N}^{\top}\|_{F}$
$\displaystyle\leq\|\widehat{\mu}_{N}-\mu\|+\frac{1}{\lambda_{\min}(\Sigma)}\|\widehat{\mu}_{N}\widehat{\mu}_{N}^{\top}-\mu\mu^{\top}\|_{F}+\frac{\sqrt{n}}{\lambda_{\min}(\Sigma)}\,\|\widehat{M}_{N}-M\|,$

where the second inequality holds because $\sqrt{a^{2}+b^{2}}\leq a+b$ for all
$a,b\geq 0$, and the third inequality follows from [96, Equation (1.2)]. The
last inequality exploits the triangle inequality and the observation that
$\|A\|_{F}\leq\sqrt{n}\|A\|$ for all $A\in\mathbb{S}^{n}$. Note that all
divisions by $\lambda_{\min}(\Sigma)$ are well-defined because $\Sigma\succ 0$
by assumption. An elementary calculation further shows that

$\displaystyle\|\widehat{\mu}_{N}\widehat{\mu}_{N}^{\top}-\mu\mu^{\top}\|_{F}$
$\displaystyle=\|\widehat{\mu}_{N}(\widehat{\mu}_{N}-\mu)^{\top}+(\widehat{\mu}_{N}-\mu)\mu^{\top}\|_{F}$
$\displaystyle\leq\|\widehat{\mu}_{N}(\widehat{\mu}_{N}-\mu)^{\top}\|_{F}+\|(\widehat{\mu}_{N}-\mu)\mu^{\top}\|_{F}$
$\displaystyle\leq\|\widehat{\mu}_{N}\|\cdot\|\widehat{\mu}_{N}-\mu\|+\|\widehat{\mu}_{N}-\mu\|\cdot\|\mu\|$
$\displaystyle=\|\widehat{\mu}_{N}-\mu\|\,(\|\widehat{\mu}_{N}\|+\|\mu\|)$
$\displaystyle\leq
2\|\mu\|\cdot\|\widehat{\mu}_{N}-\mu\|+\|\widehat{\mu}_{N}-\mu\|^{2}.$

Next, set $K=1+2\|\mu\|/\lambda_{\min}(\Sigma)$, and introduce an auxiliary
function $f_{\mu}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ defined through
$f_{\mu}(x)=Kx+x^{2}/\lambda_{\min}(\Sigma)$ for all $x\in\mathbb{R}_{+}$.
Similarly, introduce an auxiliary function
$f_{M}:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ defined through
$f_{M}(x)=\sqrt{n}x/\lambda_{\min}(\Sigma)$ for all $x\in\mathbb{R}_{+}$. The
above estimates imply that

$\mathds{G}\big{(}(\widehat{\mu}_{N},\widehat{\Sigma}_{N}),(\mu,\Sigma)\big{)}\leq
f_{\mu}\big{(}\|\widehat{\mu}_{N}-\mu\|\big{)}+f_{M}\big{(}\|\widehat{M}_{N}-M\|\big{)}.$

For any $\eta_{\mu},\eta_{M}\in(0,1]$ and for any $\rho\geq
f_{\mu}(\rho_{\mu}(\eta_{\mu}))+f_{M}(\rho_{M}(\eta_{M}))$, we thus have

$\displaystyle\phantom{\geq}\mathbb{P}^{N}\left[\mathds{G}\big{(}(\widehat{\mu},\widehat{\Sigma}),(\mu,\Sigma)\big{)}\leq\rho\right]$
$\displaystyle\geq\mathbb{P}^{N}\left[f_{\mu}\big{(}\|\widehat{\mu}_{N}-\mu\|\big{)}+f_{M}\big{(}\|\widehat{M}_{N}-M\|\big{)}\leq\rho\right]$
$\displaystyle\geq\mathbb{P}^{N}\left[f_{\mu}\big{(}\|\widehat{\mu}_{N}-\mu\|\big{)}\leq
f_{\mu}(\rho_{\mu}(\eta_{\mu}))~{}\wedge~{}f_{M}\big{(}\|\widehat{M}_{N}-M\|\big{)}\leq
f_{M}(\rho_{M}(\eta_{M}))\right]$ (32)
$\displaystyle=\mathbb{P}^{N}\left[\|\widehat{\mu}_{N}-\mu\|\leq\rho_{\mu}(\eta_{\mu})~{}\wedge~{}\|\widehat{M}_{N}-M\|\leq\rho_{M}(\eta_{M})\right]\geq
1-\eta_{\mu}-\eta_{M},$

where the equality holds because $f_{\mu}$ and $f_{M}$ are strictly increasing
on their domains, and the last inequality follows from the reverse union
bound. The claim thus follows if we define
$\rho(\eta_{\mu},\eta_{M})=f_{\mu}(\rho_{\mu}(\eta_{\mu}))+f_{M}(\rho_{M}(\eta_{M}))$,
which equals
$c_{1}\rho_{\mu}(\eta_{\mu})+c_{2}\rho_{\mu}(\eta_{\mu})^{2}+c_{3}\rho_{M}(\eta_{M})$
for $c_{1}=1+2\|\mu\|/\lambda_{\min}(\Sigma)$,
$c_{2}=1/\lambda_{\min}(\Sigma)$ and $c_{3}=\sqrt{n}/\lambda_{\min}(\Sigma)$.
Note that these constants are indeed strictly positive. ∎

The proof of Corollary 2 requires two preparatory lemmas.

######  Lemma 5 (Concentration inequality for the sample mean).

Suppose that $\mathbb{P}$ is sub-Gaussian with variance proxy $\sigma^{2}$,
mean $\mu$ and covariance matrix $\Sigma\succ 0$, and denote by
$\widehat{\mu}_{N}$ the sample mean corresponding to $N$ independent points
sampled from $\mathbb{P}$. Then, there exists a positive constant
$C\leq\sigma/\sqrt{\|\Sigma\|}$ such that
$\mathbb{P}^{N}[\|\widehat{\mu}_{N}-\mu\|\leq\rho_{\mu}(\eta)]\geq 1-\eta$ for
any significance level $\eta\in(0,1]$, where

$\rho_{\mu}(\eta)=C\left(\sqrt{\frac{\operatorname{Tr}\big{[}\Sigma\big{]}}{N}}+\sqrt{\frac{2\,\|\Sigma\|\log\left(1/\eta\right)}{N}}\right).$

Lemma 5 establishes a variant of the Hanson-Wright inequality [48] for sub-
Gaussian distributions.

###### Proof of Lemma 5.

Set $\tilde{\xi}_{i}=\Sigma^{-\frac{1}{2}}(\xi_{i}-\mu)$, which is well-
defined because $\Sigma\succ 0$, and note that $\tilde{\xi}_{i}$ is isotropic
in the sense that $\mathds{E}_{\mathbb{P}}[\tilde{\xi}_{i}]=0$ and
$\mathds{E}_{\mathbb{P}}[\tilde{\xi}_{i}\tilde{\xi}_{i}^{\top}]=I$ for all
$i=1,\ldots,N$. Then, the probability distribution of $\tilde{\xi}_{i}$ is
also sub-Gaussian with the variance proxy $C^{2}$ satisfying
$C^{2}\leq\sigma^{2}/\|\Sigma\|$ because

$\displaystyle\mathds{E}_{\mathbb{P}}\Big{[}\exp\big{(}z^{\top}(\tilde{\xi}_{i}-\mathds{E}_{\mathbb{P}}[\tilde{\xi}_{i}])\big{)}\Big{]}$
$\displaystyle=\mathds{E}_{\mathbb{P}}\Big{[}\exp\big{(}z^{\top}\Sigma^{-\frac{1}{2}}(\xi_{i}-\mathds{E}_{\mathbb{P}}[\xi_{i}])\big{)}\Big{]}$
$\displaystyle\leq\exp\left({\textstyle\frac{1}{2}}\|\Sigma^{-\frac{1}{2}}z\|^{2}\sigma^{2}\right)\leq\exp\left({\textstyle\frac{1}{2}}\|z\|^{2}\sigma^{2}/\|\Sigma\|\right)$

for all $z\in\mathbb{R}^{n}$. By the definition of the sample mean, we then
find

$\widehat{\mu}_{N}-\mu=\frac{1}{N}\sum_{i=1}^{N}\xi_{i}-\mu=\Sigma^{\frac{1}{2}}\left(\frac{1}{N}\sum_{i=1}^{N}\tilde{\xi}_{i}\right).$

Note that the random vector $N^{-1}\sum_{i=1}^{N}\tilde{\xi}_{i}$ has zero
mean and covariance matrix $N^{-1}I$, and thus it concentrates around $0$ for
large $N$. In addition, as $\tilde{\xi}_{1},\dots,\tilde{\xi}_{N}$ are
mutually independent, one easily verifies that
$N^{-1}\sum_{i=1}^{N}\tilde{\xi}_{i}$ is also sub-Gaussian with variance proxy
$C^{2}/N$. Therefore, [50, Theorem 2.1] guarantees that

$\displaystyle\mathbb{P}^{N}\left[\left\|\Sigma^{\frac{1}{2}}\left(\frac{1}{N}\sum_{i=1}^{N}\tilde{\xi}_{i}\right)\right\|^{2}\leq\frac{C^{2}}{N}\left(\operatorname{Tr}\big{[}\Sigma\big{]}+2\sqrt{\operatorname{Tr}\big{[}\Sigma^{2}\big{]}\log\left(1/\eta\right)}+2\|\Sigma\|\log\left(1/\eta\right)\right)\right]\geq
1-\eta.$

The elementary inequality
$\operatorname{Tr}\big{[}\Sigma^{2}\big{]}\leq\|\Sigma\|\operatorname{Tr}\big{[}\Sigma\big{]}$
for any $\Sigma\succeq 0$ further implies that

$\operatorname{Tr}\big{[}\Sigma\big{]}+2\sqrt{\operatorname{Tr}\big{[}\Sigma^{2}\big{]}\log\left(1/\eta\right)}+2\|\Sigma\|\log\left(1/\eta\right)\leq\left(\sqrt{\operatorname{Tr}\big{[}\Sigma\big{]}}+\sqrt{2\|\Sigma\|\log\left(1/\eta\right)}\right)^{2}.$

Combining this inequality with the above concentration bound yields

$\displaystyle\mathbb{P}^{N}\left[\|\widehat{\mu}_{N}-\mu\|\leq\rho_{\mu}(\eta)\right]$
$\displaystyle=\;$
$\displaystyle\mathbb{P}^{N}\left[\left\|\Sigma^{\frac{1}{2}}\left(\frac{1}{N}\sum_{i=1}^{N}\tilde{\xi}_{i}\right)\right\|\leq
C\left(\sqrt{\frac{\operatorname{Tr}\big{[}\Sigma\big{]}}{N}}+\sqrt{\frac{2\|\Sigma\|\log\left(1/\eta\right)}{N}}\right)\right]$
$\displaystyle\geq\;$
$\displaystyle\mathbb{P}^{N}\left[\left\|\Sigma^{\frac{1}{2}}\left(\frac{1}{N}\sum_{i=1}^{N}\tilde{\xi}_{i}\right)\right\|^{2}\leq\frac{C^{2}}{N}\left(\operatorname{Tr}\big{[}\Sigma\big{]}+2\sqrt{\operatorname{Tr}\big{[}\Sigma^{2}\big{]}\log\left(1/\eta\right)}+2\|\Sigma\|\log\left(1/\eta\right)\right)\right]$
$\displaystyle\geq\;$ $\displaystyle 1-\eta,$

and thus the claim follows. ∎

######  Lemma 6 (Concentration inequality for the sample second moment
matrix).

Suppose that $\mathbb{P}$ is sub-Gaussian with variance proxy $\sigma^{2}$ and
second moment matrix $M$, and denote by $\widehat{M}_{N}$ the sample second
moment matrix corresponding to $N$ independent points sampled from
$\mathbb{P}$. Then, there are universal constants $C_{1}>0$, $C_{2}\geq 1$,
$C_{3}>0$ such that
$\mathbb{P}^{N}[\|\widehat{M}_{N}-M\|\leq\rho_{M}(\eta)]\geq 1-\eta$ for any
significance level $\eta\in(0,1]$, where

$\rho_{M}(\eta)=\sigma^{2}C_{1}\left(\sqrt{\frac{n}{N}}+\frac{n}{N}\right)+\sigma^{2}\left(\sqrt{\frac{\log(C_{2}/\eta)}{C_{3}N}}+\frac{\log(C_{2}/\eta)}{C_{3}N}\right).$

###### Proof of Lemma 6.

By [104, Theorem 6.5], for any $\delta>0$ there exist universal constants
$C_{1}>0$, $C_{2}\geq 1$, $C_{3}>0$ with

$\mathbb{P}^{N}\left[\frac{\|\widehat{M}_{N}-M\|}{\sigma^{2}}>C_{1}\left(\sqrt{\frac{n}{N}}+\frac{n}{N}\right)+\delta\right]\leq
C_{2}\exp\left(-C_{3}N\min\\{\delta,\delta^{2}\\}\right),$

which implies that

$\displaystyle\mathbb{P}^{N}\\!\\!\left[\|\widehat{M}_{N}-M\|\leq\sigma^{2}C_{1}\left(\sqrt{\frac{n}{N}}+\frac{n}{N}\right)+\sigma^{2}\max\left\\{\sqrt{\frac{\log(C_{2}/\eta)}{C_{3}N}},\frac{\log(C_{2}/\eta)}{C_{3}N}\right\\}\right]\\!\geq\\!1-\eta.$

The claim then follows from the inequality $\max\\{a,b\\}\leq a+b$ for all
$a,b\geq~{}0$. ∎

We are now armed to prove Corollary 2.

###### Proof of Corollary 2.

Define $f_{\mu}$ and $f_{M}$ as in the proof of Theorem 4. Lemma 5 then
implies that

$\displaystyle
f_{\mu}(\rho_{\mu}(\eta_{\mu}))=CK\left(\sqrt{\frac{\operatorname{Tr}\big{[}\Sigma\big{]}}{N}}+\sqrt{\frac{2\|\Sigma\|\log\left(1/\eta_{\mu}\right)}{N}}\right)+C^{2}\left(\sqrt{\frac{\operatorname{Tr}\big{[}\Sigma\big{]}}{\lambda_{\min}(\Sigma)N}}+\sqrt{\frac{2\|\Sigma\|\log\left(1/\eta_{\mu}\right)}{\lambda_{\min}(\Sigma)N}}\right)^{2}$
$\displaystyle\leq\frac{CK\sqrt{\operatorname{Tr}\big{[}\Sigma\big{]}}\lambda_{\min}(\Sigma)+CK\lambda_{\min}(\Sigma)+2C^{2}\operatorname{Tr}\big{[}\Sigma\big{]}}{\lambda_{\min}(\Sigma)\sqrt{N}}+\frac{\left(2CK\|\Sigma\|\lambda_{\min}(\Sigma)+4C^{2}\|\Sigma\|\right)\log\left(1/\eta_{\mu}\right)}{\lambda_{\min}(\Sigma)\sqrt{N}},$

where the inequality holds because $(a+b)^{2}\leq 2a^{2}+2b^{2}$ for all
$a,b\geq 0$, while $1/N\leq 1/\sqrt{N}$ for all $N\in\mathbb{N}$ and
$\sqrt{x}\leq 1+x$ for all $x\geq 0$. Similarly, Lemma 6 implies that

$\displaystyle f_{M}(\rho_{M}(\eta_{M}))$
$\displaystyle=\frac{C_{1}\sigma^{2}\sqrt{n}}{\lambda_{\min}(\Sigma)}\left(\sqrt{\frac{n}{N}}+\frac{n}{N}\right)+\frac{\sigma^{2}\sqrt{n}}{\lambda_{\min}}\left(\sqrt{\frac{\log(C_{2}/\eta_{M})}{C_{3}N}}+\frac{\log(C_{2}/\eta_{M})}{C_{3}N}\right)$
$\displaystyle\leq\frac{n\sigma^{2}C_{1}C_{3}(1+\sqrt{n})+\sigma^{2}\sqrt{C_{3}n}}{\lambda_{\min}(\Sigma)C_{3}\sqrt{N}}+\frac{\sigma^{2}\sqrt{n}(\sqrt{C_{3}}+1)\log(C_{2}/\eta_{M})}{\lambda_{\min}(\Sigma)C_{3}\sqrt{N}},$

where the inequality holds again because $1/N\leq 1/\sqrt{N}$ for all
$N\in\mathbb{N}$ and $\sqrt{x}\leq 1+x$ for all $x\geq 0$. Setting
$\eta_{\mu}=\eta_{M}=\eta/2$, the sum of the above upper bounds on
$f_{\mu}(\rho_{\mu}(\eta_{\mu}))$ and $f_{M}(\rho_{M}(\eta_{M}))$ equals
$\rho(\eta)=(c_{1}+c_{2}\log(1/\eta))/\sqrt{N}$ for some positive constants
$c_{1}$ and $c_{2}$ that depend only on $\mu$, $\Sigma$, $\sigma^{2}$ and $n$.
The claim then follows from Theorem 4. ∎

###### Proof of Proposition 1.

The inclusion (4) follows immediately from the Gelbrich bound of Theorem 1. It
thus suffices to prove the reverse inclusion for $\widehat{\Sigma}\succ 0$. To
this end, select any
$(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$, and
construct the pushforward distribution $\mathbb{Q}=\widehat{\mathbb{P}}\circ
f^{-1}$ using the affine function
$f(\xi)=\widehat{\Sigma}^{-\frac{1}{2}}(\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}})^{\frac{1}{2}}\widehat{\Sigma}^{-\frac{1}{2}}(\xi-\widehat{\mu})+\mu$.
As the structural ambiguity set $\mathcal{S}$ is closed under positive
semidefinite affine pushforwards, we have $\mathbb{Q}\in\mathcal{S}$. In
addition, Theorem 2 implies that
$\mathds{W}(\mathbb{Q},\widehat{\mathbb{P}})=\mathds{G}\big{(}(\mu,\Sigma),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\rho$,
where the inequality holds because
$(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$. We may
thus conclude that $\mathbb{Q}\in\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$.
Finally, an elementary calculation reveals that $\mathbb{Q}$ has mean $\mu$
and covariance matrix $\Sigma$. ∎

###### Proof of Theorem 3.

The claim trivially holds if $\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$ is
empty. From now on we thus assume that
$\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$ is non-empty. For any distribution
$\mathbb{Q}\in\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$ with mean
$\mu\in\mathbb{R}^{n}$ and covariance matrix $\Sigma\in\mathbb{S}_{+}^{n}$,
Theorem 1 then implies that

$\mathds{G}\big{(}(\mu,\Sigma),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\mathds{W}(\mathbb{Q},\widehat{\mathbb{P}})\leq\rho.$

As $\mathbb{Q}\in\mathcal{S}$, we thus have
$\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$. Hence, we
may conclude that
$\mathcal{W}_{\rho}(\widehat{\mathbb{P}})\subseteq\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$.

Assume now that the structural ambiguity set $\mathcal{S}$ is generated by
$\widehat{\mathbb{P}}$ and that $\widehat{\Sigma}\succ 0$. Next, select any
distribution $\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$
with mean $\mu\in\mathbb{R}^{n}$ and covariance matrix
$\Sigma\in\mathbb{S}_{+}^{n}$. As $\mathbb{Q}\in\mathcal{S}$ constitutes a
positive semidefinite affine pushforward of $\widehat{\mathbb{P}}$, we thus
have

$\mathds{W}(\mathbb{Q},\widehat{\mathbb{P}})=\mathds{G}\big{(}(\mu,\Sigma),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\rho,$

where the equality follows from Theorem 2. This implies that
$\mathbb{Q}\in\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$ and, as
$\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ was chosen
arbitrarily, that
$\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})\subseteq\mathcal{W}_{\rho}(\widehat{\mathbb{P}})$.
Recalling the first part of the proof, we then obtain
$\mathcal{W}_{\rho}(\widehat{\mathbb{P}})=\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$.
∎

###### Proof of Proposition 2.

The non-negativity of the Gelbrich distance implies that

$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})=\left\\{(\mu,\Sigma)\in\mathbb{R}^{n}\times\mathbb{S}_{+}^{n}:\mathds{G}^{2}\big{(}(\mu,\Sigma),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\rho^{2}\right\\}.$

Recall that the squared Gelbrich distance is convex in
$(\mu,\Sigma)\in\mathbb{R}^{n}\times\mathbb{S}_{+}^{n}$. The Hölder continuity
of the matrix square root established in [79, Lemma A.2] ensures that the
squared Gelbrich distance is also continuous. Therefore,
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is convex and closed.
Finally, one can show that for any
$(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ we have
$\|\mu-\widehat{\mu}\|\leq\rho$ and
$0\preceq\Sigma\preceq\big{(}\rho+\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}^{\frac{1}{2}}\big{)}^{2}I$,
see [97, Lemma A.6]. This implies that
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is also compact. ∎

###### Proof of Proposition 3.

Recall from Proposition 2 that
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is compact, and note that
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ can be viewed as the
image of $\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ under the
continuous transformation $f(\mu,\Sigma)=(\mu,\Sigma+\mu\mu^{\top})$ defined
on $\mathbb{R}^{n}\times\mathbb{S}_{+}^{n}$. Thus, the transformed set
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ inherits compactness from
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$. To show that
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is convex, recall from
the proof of Theorem 1 that the squared Gelbrich distance satisfies

$\displaystyle\mathds{G}^{2}\big{(}(\mu,\Sigma),(\widehat{\mu},\widehat{\Sigma})\big{)}$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle\inf&\displaystyle\|\mu-\widehat{\mu}\|^{2}+\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2C\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\displaystyle C\in\mathbb{R}^{n\times
n},\;\begin{bmatrix}\Sigma&C\\\ C^{\top}&\widehat{\Sigma}\end{bmatrix}\succeq
0\end{array}\right.$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle\sup&\|\mu-\widehat{\mu}\|^{2}+\displaystyle\operatorname{Tr}\big{[}\Sigma(I-A_{11})+\widehat{\Sigma}(I-A_{22})\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\displaystyle
A_{11}\in\mathbb{S}_{+}^{n},\;A_{22}\in\mathbb{S}_{+}^{n},\;\begin{bmatrix}A_{11}&-I\\\
-I&A_{22}\end{bmatrix}\succeq 0.\end{array}\right.$

Here, the second equality follows from strong semidefinite duality [5, Theorem
1.4.2], which holds because $A_{11}=A_{22}=2I$ constitutes a Slater point for
the dual problem. Thus, we have

$\displaystyle\phantom{=}\mathds{G}^{2}\big{(}(\mu,M-\mu\mu^{\top}),(\widehat{\mu},\widehat{\Sigma})\big{)}$
$\displaystyle=\left\\{\begin{array}[]{cl}\displaystyle\sup&\displaystyle\operatorname{Tr}\big{[}M(I-A_{22})\big{]}+\mu^{\top}A_{22}\mu-2\mu^{\top}\widehat{\mu}+\operatorname{Tr}\big{[}\widehat{\Sigma}(I-A_{22})\big{]}+\|\widehat{\mu}\|^{2}\\\\[4.30554pt]
\operatorname{s.t.}&\displaystyle
A_{11}\in\mathbb{S}_{+}^{n},\;A_{22}\in\mathbb{S}_{+}^{n},\;\begin{bmatrix}A_{11}&-I\\\
-I&A_{22}\end{bmatrix}\succeq 0\end{array}\right.$

for any $\mu\in\mathbb{R}^{n}$ and $M\in\mathbb{S}_{+}^{n}$ with
$M\succeq\mu\mu^{\top}$. Note that the objective function of the above
maximization problem is jointly convex in $\mu$ and $M$ for any feasible
$A_{11}$ and $A_{22}$. As convexity is preserved under maximization, we may
thus conclude that
$\mathds{G}^{2}((\mu,M-\mu\mu^{\top}),(\widehat{\mu},\widehat{\Sigma}))$ is
also jointly convex in $\mu$ and $M$. Hence, the set

$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})=\left\\{(\mu,M)\in\mathbb{R}^{n}\times\mathbb{S}_{+}^{n}:M\succeq\mu\mu^{\top},~{}\mathds{G}^{2}\big{(}(\mu,M-\mu\mu^{\top}),(\widehat{\mu},\widehat{\Sigma})\big{)}\leq\rho^{2}\right\\}$

is convex because it is representable as the feasible set of convex
constraints. ∎

##  Appendix C Proofs of Section 4

###### Proof of Proposition 4.

Define $\mathcal{F}$ as the set of all cumulative distribution functions on
$\mathbb{R}$. Thus, $\mathcal{F}$ contains all non-decreasing and right-
continuous functions $F:\mathbb{R}\rightarrow[0,1]$ with
$\lim_{t\downarrow\infty}F(t)=0$ and $\lim_{t\uparrow\infty}F(t)=1$. For any
loss function $\ell\in\mathcal{L}_{0}$ and probability distribution
$\mathbb{Q}\in\mathcal{M}$, we use $F_{\ell(\xi)}^{\mathbb{Q}}\in\mathcal{F}$
to denote the distribution function of the random variable $\ell(\xi)$ under
$\mathbb{Q}$. As the family of risk measures
$\\{\mathcal{R}_{\mathbb{Q}}\\}_{\mathbb{Q}\in\mathcal{M}}$ is law-invariant,
there exists a distribution functional $\varrho:\mathcal{F}\to\mathbb{R}$ with
$\mathcal{R}_{\mathbb{Q}}(\ell)=\varrho(F_{\ell(\xi)}^{\mathbb{Q}})$ for all
$\ell\in\mathcal{L}_{0}$ and $\mathbb{Q}\in\mathcal{M}$.

Define now $\mathcal{F}(0,1)\subseteq\mathcal{F}$ as the family of all
cumulative distribution functions of the random variables of the form
$v^{\top}\xi$ for some $v\in\mathbb{R}^{n}$ under any probability distribution
$\mathbb{D}\in\mathcal{S}$ under which $v^{\top}\xi$ has zero mean and unit
variance, that is, we set

$\mathcal{F}(0,1)=\left\\{F^{\mathbb{D}}_{v^{\top}\xi}:v\in\mathbb{R}^{n},~{}\mathbb{D}\in\mathcal{S}\text{
such that }\mathds{E}_{\mathbb{D}}[v^{\top}\xi]=0\text{ and
}\mathds{E}_{\mathbb{D}}[(v^{\top}\xi)^{2}]=1\right\\}.$

Next, choose any $\mu\in\mathbb{R}^{n}$, $\Sigma\in\mathbb{S}_{+}^{n}$ and
$w\in\mathbb{R}^{n}$ with $w^{\top}\Sigma w>0$, and set $m=w^{\top}\mu$ and
$s=\sqrt{w^{\top}\Sigma w}$. Recalling that the structured Chebyshev ambiguity
set $\mathcal{C}(\mu,\Sigma)$ contains all distributions
$\mathbb{Q}\in\mathcal{S}$ with mean $\mu$ and covariance matrix $\Sigma$, we
will first show that

$\mathcal{F}(0,1)=\left\\{F_{-(w^{\top}\xi-m)/s}^{\mathbb{Q}}:\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)\right\\}.$
(33)

The proof proceeds in two steps. In the first step, we prove that the left
hand side of (33) is a subset of the right hand side. To this end, select any
$F\in\mathcal{F}(0,1)$. Thus, there exists a random vector
$\xi_{1}\in\mathbb{R}^{n}$ with probability distribution
$\mathbb{D}_{1}\in\mathcal{S}$ and a deterministic vector $v\in\mathbb{R}^{n}$
such that $v^{\top}\xi_{1}$ has zero mean and unit variance under
$\mathbb{D}_{1}$ and such that $F=F^{\mathbb{D}_{1}}_{v^{\top}\xi_{1}}$. Using
similar ideas as in [108, Theorems 1 and 2], we construct another random
vector $\xi_{2}\in\mathbb{R}^{n}$ with distribution
$\mathbb{D}_{2}\in\mathcal{C}(0,I)$ that is independent of $\xi_{1}$. We can
then construct a new random vector $\xi\in\mathbb{R}^{n}$ as the following
linear combination of $\xi_{1}$ and $\xi_{2}$.

$\displaystyle\textstyle\xi=\mu-\frac{1}{s}\Sigma
wv^{\top}\xi_{1}+\left(I-\frac{1}{s^{2}}\Sigma
ww^{\top}\right)\Sigma^{\frac{1}{2}}\xi_{2}$

Next, we define $\mathbb{Q}$ as the probability distribution of $\xi$. As
$\mathbb{D}_{1}$ and $\mathbb{D}_{2}$ belong to the structural ambiguity set
$\mathcal{S}$ and as $\mathcal{S}$ is stable and therefore closed under (not
necessarily positive semidefinite) affine pushforwards and under convolutions,
we may conclude that $\mathbb{Q}\in\mathcal{S}$. In addition, an elementary
calculation exploiting our knowledge that $v^{\top}\xi_{1}$ and $\xi_{2}$ are
standardized under the distributions $\mathbb{D}_{1}$ and $\mathbb{D}_{2}$,
respectively, reveals that $\xi$ has mean $\mu$ and covariance matrix $\Sigma$
under $\mathbb{Q}$. Hence, we have shown that
$\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)$. By the construction of $\xi$ and the
definitions of $m$ and $s$, we finally have
$-(w^{\top}\xi-m)/s=v^{\top}\xi_{1}$ and thus
$F_{-(w^{\top}\xi-m)/s}^{\mathbb{Q}}=F$. This implies that $F$ belongs to the
set on the right hand side of (33).

In the second step, we prove that the right hand side of (33) is a subset of
the left hand side. To this end, select any
$\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)$, and use $F$ as a shorthand for
$F^{\mathbb{Q}}_{-(w^{\top}\xi-m)/s}$. To show that $F\in\mathcal{F}(0,1)$,
set $v=-w/s$, and define the pushforward distribution
$\mathbb{D}=\mathbb{Q}\circ f^{-1}$ with respect to the transformation
$f(\xi)=\xi+mv/(s\|v\|^{2})$. Note first that $\mathbb{D}\in\mathcal{S}$
because the structural ambiguity set $\mathcal{S}$ is closed under affine
pushforwards. In addition, one readily verifies that
$-(w^{\top}\xi-m)/s=v^{\top}f(\xi)$ has zero mean and unit variance under
$\mathbb{Q}$, which means that $v^{\top}\xi$ has zero mean and unit variance
under $\mathbb{D}$. We thus have $F\in\mathcal{F}(0,1)$. In combination, the
first and the second step of the proof establish equation (33).

We may now conclude that the standard risk coefficient satisfies

$\displaystyle\alpha=\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\;\mathcal{R}_{\mathbb{Q}}\left(-\frac{1}{s}(w^{\top}\xi-m)\right)=\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\;\varrho\left(F^{\mathbb{Q}}_{-(w^{\top}\xi-m)/s}\right)=\sup\limits_{F\in\mathcal{F}(0,1)}\;\varrho(F),$

where the second equality re-expresses the risk measures
$\mathcal{R}_{\mathbb{Q}}$ for $\mathbb{Q}\in\mathcal{M}$ in terms of the
distribution functional $\varrho$, and the second equality exploits (33). The
last expression is manifestly independent of $\mu$, $\Sigma$ and $w$. This
observation completes the proof. ∎

###### Proof of Theorem 5.

Decomposing the Gelbrich ambiguity set into disjoint Chebyshev ambiguity sets
allows us to rewrite the Gelbrich risk as in (6a). As the loss function
$\ell(\xi)=-w^{\top}\xi$ is linear, the inner maximization problem in (6a)
further simplifies to

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\;\mathcal{R}_{\mathbb{Q}}\left(-w^{\top}\xi\right)=-w^{\top}\mu+\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\;\mathcal{R}_{\mathbb{Q}}\left(-w^{\top}(\xi-\mu)\right)$
$\displaystyle\qquad=-w^{\top}\mu+\sqrt{w^{\top}\Sigma
w}~{}\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\;\mathcal{R}_{\mathbb{Q}}\left(-\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma
w}}\right)=-w^{\top}\mu+\alpha\sqrt{w^{\top}\Sigma w}$

where the first two equalities exploit the translation invariance and the
positive homogeneity of the risk measures $\mathcal{R}_{\mathbb{Q}}$,
$\mathbb{Q}\in\mathcal{M}$, respectively, whereas the third equality follows
from the definition of the standard risk coefficient $\alpha$. Note that
$\alpha$ is independent of $\mu$, $\Sigma$ and $w$ thanks to Proposition 4,
which applies because the structural ambiguity set is stable and the family of
risk measures is law-invariant. Using the definition of the set
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$, the outer maximization
problem in (6a) can then be reformulated as

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;\mathcal{R}_{\mathbb{Q}}(-w^{\top}\xi)$
$\displaystyle=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;-\mu^{\top}w+\alpha\sqrt{w^{\top}\Sigma
w}$ $\displaystyle=\left\\{\begin{array}[]{cl}\sup\limits_{\mu,\Sigma\succeq
0}&-\mu^{\top}w+\alpha\sqrt{w^{\top}\Sigma w}\\\
\operatorname{s.t.}&\|\mu-\widehat{\mu}\|^{2}+\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}\leq\rho^{2}.\end{array}\right.$
(36)

If $\rho=0$, then $(\mu,\Sigma)=(\widehat{\mu},\widehat{\Sigma})$ is the only
feasible solution of problem (36), in which case (8) trivially holds.
Similarly, if $\alpha=0$, then $\Sigma=\widehat{\Sigma}$ and
$\mu=\widehat{\mu}-\rho w/\|w\|$ are optimal in (36), and (8) again trivially
holds. From now on we may thus assume without loss of generality that $\rho>0$
and $\alpha>0$. In this case problem (36) is equivalent to

$\displaystyle\sup\limits_{\mu,\Sigma\succeq 0}\inf\limits_{\gamma\geq
0}\quad-\mu^{\top}w+\alpha\sqrt{w^{\top}\Sigma
w}+\gamma\left[\rho^{2}-\|\mu-\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\Sigma+\widehat{\Sigma}-2\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}\right]$
$\displaystyle\quad=\inf\limits_{\gamma\geq
0}\Bigg{\\{}\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\sup\limits_{\mu}\left\\{-\mu^{\top}w-\gamma\|\mu-\widehat{\mu}\|^{2}\right\\}$
$\displaystyle\quad\hskip 30.00005pt+\sup\limits_{\Sigma\succeq
0}\left\\{\alpha\sqrt{w^{\top}\Sigma
w}+\gamma\operatorname{Tr}\big{[}-\Sigma+2\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}\right\\}\Bigg{\\}},$

where the first equality follows from strong duality, which holds because
$(\widehat{\mu},\widehat{\Sigma})$ constitutes a Slater point for the primal
convex program (36), and from a simple rearrangement. As $\gamma\geq 0$, the
embedded quadratic maximization problem over $\mu$ is convex and can be solved
analytically. Indeed, for $\gamma>0$ we have
$\sup_{\mu}\left\\{-\mu^{\top}w-\gamma\|\mu-\widehat{\mu}\|^{2}\right\\}=-\widehat{\mu}^{\top}w+\frac{\|w\|^{2}}{4\gamma}$.
For $\gamma=0$, on the other hand, the supremum over $\mu$ evaluates to $0$ if
$w=0$ and to $+\infty$ otherwise. Thus, the formula on the right hand side of
the above expression remains valid for $\gamma=0$ if we interpret it as the
limit when $\gamma$ tends to 0 from above. Similarly, as $\gamma\geq 0$ and
$\alpha>0$, one may introduce an auxiliarly epigraphical variable $t$ to
reformulate the embedded maximization problem over $\Sigma$ as the convex
program

$\begin{array}[]{cl}\sup\limits_{t,\Sigma}&~{}\alpha
t+\gamma\operatorname{Tr}\big{[}-\Sigma+2\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}\\\
\operatorname{s.t.}&~{}t\geq 0,\;\Sigma\succeq 0,\;t^{2}-w^{\top}\Sigma w\leq
0,\end{array}$

which manifestly satisfies Slater’s condition. Suppose now temporarily that
$\widehat{\Sigma}\succ 0$. By invoking strong duality and using the variable
transformation
$B\leftarrow(\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}})^{\frac{1}{2}}$,
the above optimization problem can be recast as

$\displaystyle\inf\limits_{\lambda\geq 0}\sup\limits_{t\geq 0,\Sigma\succeq
0}~{}\alpha t-\lambda t^{2}+\operatorname{Tr}\big{[}\Sigma(\lambda
ww^{\top}-\gamma
I)\big{]}+2\gamma\operatorname{Tr}\big{[}\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}$
$\displaystyle=$ $\displaystyle\inf\limits_{\lambda\geq 0}\sup\limits_{t\geq
0,B\succeq 0}~{}\alpha t-\lambda
t^{2}+\operatorname{Tr}\big{[}B^{2}\Delta_{\lambda}\big{]}+2\gamma\operatorname{Tr}\big{[}B\big{]},$
(37)

where $\Delta_{\lambda}=\widehat{\Sigma}^{-\frac{1}{2}}(\lambda
ww^{\top}-\gamma I)\widehat{\Sigma}^{-\frac{1}{2}}$ for any $\lambda\geq 0$.
Note that $\Delta_{\lambda}$ is well-defined because $\widehat{\Sigma}$ is
invertible. Note also that the inner maximization problem in (37) is separable
with respect to $t$ and $B$. Consider first the maximization problem over $t$.
As $\alpha>0$, its supremum evaluates to $\alpha^{2}/(4\lambda)$ and is
attained at $t^{\star}=\alpha/(2\lambda)$ whenever $\lambda>0$. Otherwise, if
$\lambda=0$, then its supremum evaluates to $+\infty$ for. From now on we may
thus assume without loss of generality that the outer minimization over
$\lambda$ in (37) is subject to the strict constraint $\lambda>0$. Consider
now the maximization problem over $B$. The proof of [78, Proposition 2.8]
implies that if $\Delta_{\lambda}\not\prec 0$, then the supremum over $B$
evaluates to $+\infty$. From now on we may thus assume without loss of
generality that the outer minimization over $\lambda$ in (37) is subject to
the strict constraint $\gamma I-\lambda ww^{\top}\succ 0$, which is equivalent
to $\lambda<\gamma\|w\|^{-2}$ and guarantees that $\Delta_{\lambda}\prec 0$.
As $\lambda>0$, this in turn implies that
$B^{\star}=-\gamma\Delta_{\lambda}^{-1}$ is strictly positive definite and
satisfies the first-order optimality condition
$B\Delta_{\lambda}+\Delta_{\lambda}B+2\gamma I=0$. This condition can be
interpreted as a continuous Lyapunov equation, and therefore its solution
$B^{\star}$ is in fact unique; see, e.g., [49, Theorem 12.5]. By making the
implicit constraints on $\lambda$ explicit and by eliminating the supremum
operator by evaluating the objective function at $t^{\star}$ and $B^{\star}$,
problem (37) can now be reformulated as

$\displaystyle\inf\limits_{0<\lambda<\gamma\|w\|^{-2}}\;\frac{\alpha^{2}}{4\lambda}+\gamma^{2}\operatorname{Tr}\big{[}\widehat{\Sigma}^{\frac{1}{2}}(\gamma
I-\lambda ww^{\top})^{-1}\widehat{\Sigma}^{\frac{1}{2}}\big{]}$
$\displaystyle=\inf\limits_{0<\lambda<\gamma\|w\|^{-2}}\;\frac{\alpha^{2}}{4\lambda}+\gamma\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+\frac{w^{\top}\widehat{\Sigma}w}{\lambda^{-1}-\|w\|^{2}/\gamma}=\gamma\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}+\frac{\alpha^{2}}{4}\frac{\|w\|^{2}}{\gamma}+\alpha\sqrt{w^{\top}\widehat{\Sigma}w}.$

Here, the first equality exploits the Sherman-Morrison formula [7, Corollary
2.8.8] to rewrite the inverse matrix, and the second equality solves the
resulting minimization problem analytically. Indeed, the infimum is attained
in the interior of the feasible set at the unique solution $\lambda^{\star}$
of the first-order condition
$\frac{1}{\lambda}=\frac{\|w\|^{2}}{\gamma}+\frac{2}{\alpha}\sqrt{w^{\top}\widehat{\Sigma}w}$.
In summary, we have derived closed-form solutions for both subproblems over
$\mu$ and $\Sigma$ in the objective function of the problem dual to (36).
Hence, the Gelbrich risk satisfies

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;\mathcal{R}_{\mathbb{Q}}(-w^{\top}\xi)$
$\displaystyle=\inf\limits_{\gamma\geq
0}~{}-\widehat{\mu}^{\top}w+\alpha\sqrt{w^{\top}\widehat{\Sigma}w}+\gamma\rho^{2}+\frac{1+\alpha^{2}}{4}\frac{\|w\|^{2}}{\gamma}$
$\displaystyle=-\widehat{\mu}^{\top}w+\alpha\sqrt{w^{\top}\widehat{\Sigma}w}+\rho\sqrt{1+\alpha^{2}}\,\|w\|,$

where the second equality holds because the unique solution of the
minimization problem in the first line is given by
$\gamma^{\star}=(2\rho)^{-1}\sqrt{1+\alpha^{2}}\,\|w\|$. We have thus
established (8) for $\widehat{\Sigma}\succ 0$.

To demonstrate that (8) remains valid for all $\widehat{\Sigma}\succeq 0$, we
denote by $J(\widehat{\Sigma})$ the optimal value of problem (36) as a
function of $\widehat{\Sigma}$. Thus, $J(\widehat{\Sigma})$ coincides with the
Gelbrich risk.

Applying Berge’s maximum theorem [6, pp. 115–116] to problem (36), it is easy
to show that $J(\widehat{\Sigma})$ is continuous on $\mathbb{S}_{+}^{n}$.
Next, define
$\bar{J}(\widehat{\Sigma})=-\widehat{\mu}^{\top}w+\alpha\sqrt{w^{\top}\widehat{\Sigma}w}+\rho\sqrt{1+\alpha^{2}}\,\|w\|$
as the right hand side of (8), which is manifestly continuous on
$\mathbb{S}_{+}^{n}$. From the first part of the proof we know that
$J(\widehat{\Sigma})=\bar{J}(\widehat{\Sigma})$ for all $\widehat{\Sigma}\succ
0$. As both $J(\widehat{\Sigma})$ and $\bar{J}(\widehat{\Sigma})$ are
continuous on $\mathbb{S}_{+}^{n}$ and as any positive semidefinite matrix can
be expressed as a limit of positive definite matrices, we thus have
$J(\widehat{\Sigma})=\bar{J}(\widehat{\Sigma})$ for all
$\widehat{\Sigma}\in\mathbb{S}_{+}^{n}$. This proves (8) for all
$\widehat{\Sigma}\succeq 0$.

Finally, if $\mathcal{S}$ is the structural ambiguity set generated by a
Gaussian nominal distribution $\widehat{\mathbb{P}}$ with
$\widehat{\Sigma}\succ 0$, then the Wasserstein risk equals the Gelbrich risk
by Corollary 1. ∎

###### Proof of Proposition 5.

Fix any $\mu\in\mathbb{R}^{n}$, $\Sigma\in\mathbb{S}_{+}^{n}$ and
$w\in\mathbb{R}^{n}$, and select any symmetric probability distribution
$\mathbb{Q}\in\mathcal{S}$, which exists by assumption. As $\mathcal{S}$ is
closed under positive semidefinite affine pushforwards, we may assume without
loss of generality that $\xi$ has mean $\mu$ and covariance matrix $\Sigma$
under $\mathbb{Q}$. We then find

$\displaystyle 0$
$\displaystyle=\textstyle\mathcal{R}_{\mathbb{Q}}(0)=\mathcal{R}_{\mathbb{Q}}\left(\frac{1}{2}\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma
w}}-\frac{1}{2}\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma w}}\right)$
$\displaystyle\leq\frac{1}{2}\mathcal{R}_{\mathbb{Q}}\left(\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma
w}}\right)+\frac{1}{2}\mathcal{R}_{\mathbb{Q}}\left(-\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma
w}}\right)=\mathcal{R}_{\mathbb{Q}}\left(-\frac{w^{\top}(\xi-\mu)}{\sqrt{w^{\top}\Sigma
w}}\right)\leq\alpha,$

where the first equality and the first inequality follow from the positive
homogeneity and convexity of the coherent risk measure
$\mathcal{R}_{\mathbb{Q}}$, respectively. The last equality exploits the law-
invariance of $\mathcal{R}_{\mathbb{Q}}$ and the symmetry of $\mathbb{Q}$,
which implies that $w^{\top}(\xi-\mu)/\sqrt{w^{\top}\Sigma w}$ and
$-w^{\top}(\xi-\mu)/\sqrt{w^{\top}\Sigma w}$ have the same distribution under
$\mathbb{Q}$. Finally, the last inequality follows from the definition of
$\alpha$ and the observation that $\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)$. ∎

###### Proof of Corollary 4.

Note that problem (9) has a unique minimizer for every $\rho>0$ because its
feasible set $\Omega$ is closed and its objective function is strictly convex
and coercive on $\Omega$. This minimizer coincides with the unique optimal
solution $w^{\star}(\lambda)$ of

$\min\limits_{w\in\Omega}~{}-\lambda\widehat{\mu}^{\top}w+\lambda\alpha\sqrt{w^{\top}\widehat{\Sigma}w}+\sqrt{1+\alpha^{2}}\|w\|,$

where $\lambda=1/\rho$. By Berge’s maximum theorem [6, pp. 115–116], the
function $w^{\star}(\lambda)$ is continuous on $[0,1]$, and therefore
$w^{\star}(\lambda)$ converges to $w^{\star}(0)$ as $\lambda$ tends to $0$. As
$\frac{1}{n}e\in\Omega$, however, one readily verifies that
$w^{\star}(0)=\frac{1}{n}e$. As $\lambda=1/\rho$, this reasoning shows that
the unique minimizer of (9) converges to $\frac{1}{n}e$ as $\rho$ tends to
$\infty$. ∎

###### Proof of Proposition 6.

If $\rho=0$, then all distributions in the Gelbrich ambiguity set, and in
particular all maximizers that attain the Gelbrich risk, have mean
$\widehat{\mu}$ and covariance matrix $\widehat{\Sigma}$. The claim then
follows because, for $\rho=0$, the formulas for $\mu^{\star}$ and
$\Sigma^{\star}$ reduce to $\widehat{\mu}$ and $\widehat{\Sigma}$,
respectively. Assume from now on that $\rho>0$. As all assumptions of Theorem
5 are satisfied, we may proceed as in the proof of Theorem 5 to show that

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;\mathcal{R}_{\mathbb{Q}}\left(-w^{\top}\xi\right)$
$\displaystyle=\left\\{\begin{array}[]{cl}\max&-\mu^{\top}w+\alpha t\\\
\text{s.t.}&(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma}),~{}t\geq
0,~{}t^{2}\leq w^{\top}\Sigma w\end{array}\right.$
$\displaystyle=\min\limits_{\gamma\geq 0,\lambda\geq
0}\sup\limits_{\mu,\Sigma\succeq 0,t\geq 0}L(\mu,\Sigma,t,\gamma,\lambda),$

where the last equality follows from strong duality, which applies because the
primal problem has a compact feasible set. The Lagrangian in the last
expression is defined through

$\displaystyle L(\mu,\Sigma,t,\gamma,\lambda)=\;$
$\displaystyle\gamma\big{(}\rho^{2}-\|\mu-\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}-\mu^{\top}w+\alpha
t-\lambda t^{2}$ $\displaystyle+\operatorname{Tr}\big{[}\Sigma(\lambda
ww^{\top}-\gamma
I)\big{]}+2\gamma\operatorname{Tr}\big{[}\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}.$

From the proof of Theorem 5 we know that the dual problem is uniquely solved
by

$\gamma^{\star}=(2\rho)^{-1}\sqrt{1+\alpha^{2}}\,\|w\|\quad\text{and}\quad\frac{1}{\lambda^{\star}}=\frac{\|w\|^{2}}{\gamma^{\star}}+\frac{2}{\alpha}\sqrt{w^{\top}\widehat{\Sigma}w}.$

In addition, [5, Theorem D.4.1] implies that any primal maximizer
$(\mu^{\star},\Sigma^{\star},t^{\star})$ must also be a maximizer of

$\displaystyle\max_{\mu,\Sigma\succeq 0,t\geq
0}~{}L(\mu,\Sigma,t,\gamma^{\star},\lambda^{\star}).$ (38)

As the Lagrangian is additively separable in $\mu$, $\Sigma$ and $t$, the
maximizers $\mu^{\star}$, $\Sigma^{\star}$ and $t^{\star}$ can be determined
separately. Indeed, one readily verifies that $\mu^{\star}$ must be a
maximizer of the problem
$\max_{\mu}\\{-\mu^{\top}w-\gamma^{\star}\|\mu-\widehat{\mu}\|^{2}\\}$, which
is uniquely solved by $\mu^{\star}=\widehat{\mu}-w/(2\gamma^{\star})$.
Similarly, $\Sigma^{\star}$ must be a maximizer of the problem

$\displaystyle\max\limits_{\Sigma\succeq
0}\operatorname{Tr}\big{[}\Sigma(\lambda ww^{\top}-\gamma
I)\big{]}+2\gamma\operatorname{Tr}\big{[}\big{(}\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}}\big{)}^{\frac{1}{2}}\big{]}=\max\limits_{B\succeq
0}\operatorname{Tr}\big{[}B^{2}\Delta\big{]}+2\gamma\operatorname{Tr}\big{[}B\big{]},$
(39)

where the equality exploits the substitution
$B\leftarrow(\widehat{\Sigma}^{\frac{1}{2}}\Sigma\widehat{\Sigma}^{\frac{1}{2}})^{\frac{1}{2}}$
and the definition
$\Delta=\widehat{\Sigma}^{-\frac{1}{2}}(\lambda^{\star}ww^{\top}-\gamma^{\star}I)\widehat{\Sigma}^{-\frac{1}{2}}$.
As in the proof of Theorem 5, one can show that the second maximization
problem in (39) is uniquely solved by
$B^{\star}=\widehat{\Sigma}^{\frac{1}{2}}(I-\frac{\lambda^{\star}}{\gamma^{\star}}ww^{\top})^{-1}\widehat{\Sigma}^{\frac{1}{2}}$,
which implies that the first maximization problem in (39) is uniquely solved
by

$\displaystyle\Sigma^{\star}$
$\displaystyle=\left(I-\frac{\lambda^{\star}}{\gamma^{\star}}ww^{\top}\right)^{-1}\widehat{\Sigma}\left(I-\frac{\lambda^{\star}}{\gamma^{\star}}ww^{\top}\right)^{-1}$
$\displaystyle=\left(I+\frac{\lambda^{\star}}{\gamma^{\star}-\lambda^{\star}\|w\|^{2}}ww^{\top}\right)\widehat{\Sigma}\left(I+\frac{\lambda^{\star}}{\gamma^{\star}-\lambda^{\star}\|w\|^{2}}ww^{\top}\right).$

Here, the last equality exploits the Sherman-Morrison-Woodbury identity [7,
Corollary 2.8.8]. Finally, $t^{\star}$ must be a maximizer of $\max_{t\geq
0}\alpha t-\lambda^{\star}t^{2}$, which is uniquely solved by
$t^{\star}=\alpha/(2\lambda^{\star})$. The claim now follows from the formulas
for the dual variables $\gamma^{\star}$ and $\lambda^{\star}$ substituted into
the formulas for $\mu^{\star}$, $\Sigma^{\star}$ and $t^{\star}$. ∎

###### Proof of Proposition 7.

Proposition 1 in [108] provides an analytical formula for the Chebyshev risk
of a portfolio loss function with respect to VaR. As the Chebyshev risk
coincides with the Gelbrich risk for $\rho=0$, the standard risk coefficient
is readily found by comparison with (8). ∎

###### Proof of Proposition 8.

This follows from [108, Proposition 2], similar to Proposition 7. ∎

###### Proof of Proposition 9.

Equation (8) for $\rho=0$ reveals that $\alpha=\beta$. ∎

###### Proof of Proposition 10.

This follows from [62, Theorem 2], similar to Proposition 7. ∎

###### Proof of Proposition 11.

This follows from [62, Theorem 3], similar to Proposition 7. ∎

###### Proof of Proposition 12.

This follows from [15, Theorem 3.10], similar to Proposition 7. The same
theorem reveals that the Chebyshev risk does not change if $h$ is replaced
with its right-continuous modification. In light of (6), this invariance
remains valid if uncertainty is modeled by a Gelbrich ambiguity set. Thus, we
may always assume without loss of generality that $h$ is right-continuous. ∎

##  Appendix D Proofs of Section 5

###### Proof of Theorem 6.

By the two-layer decomposition (6a) of the Gelbrich ambiguity set, we find

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathds{E}_{\mathbb{Q}}\left[\ell(\xi)\right]$
$\displaystyle=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\sup\limits_{\mathbb{Q}\in\mathcal{P}(\mu,\Sigma)}\mathds{E}_{\mathbb{Q}}[\ell(\xi)]$
(40a)
$\displaystyle=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\inf\limits_{(y_{0},y,Y)\in\mathcal{Y}}\;y_{0}+2\mu^{\top}y+\operatorname{Tr}\big{[}(\Sigma+\mu\mu^{\top})Y\big{]}$
(40b)
$\displaystyle=\sup\limits_{(\mu,M)\in\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\inf\limits_{(y_{0},y,Y)\in\mathcal{Y}}\;y_{0}+2\mu^{\top}y+\operatorname{Tr}\big{[}MY\big{]}$
(40c)
$\displaystyle=\inf\limits_{(y_{0},y,Y)\in\mathcal{Y}}\sup\limits_{(\mu,M)\in\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;y_{0}+2\mu^{\top}y+\operatorname{Tr}\big{[}MY\big{]}$
(40d)
$\displaystyle=\inf\limits_{(y_{0},y,Y)\in\mathcal{Y}}\;y_{0}+\delta^{*}_{\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(2y,Y),$
(40e)

where (40b) exploits duality. Note that the dual problem in (40b) can be
constructed as in the proof of [16, Lemma 1]. Strong duality holds because
$\rho>0$, which implies that the uncertainty set
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ contains a point
$(\mu,\Sigma)$ with $\Sigma\succ 0$. Similar arguments were used in [74, §
2.5], and thus details are omitted. In addition, (40c) holds because
$(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ if and
only if
$(\mu,\Sigma+\mu\mu^{\top})\in\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$,
(40d) follows from Sion’s minimax theorem [99], which applies because
$\mathcal{V}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ is convex and compact by
virtue of Proposition 3, and (40e) exploits the definition of the support
function. By Proposition 14, we thus obtain the optimization problem given in
the theorem statement. ∎

The proof of Theorem 7 relies on the following corollary, which generalizes
[111, Lemma A.2].

######  Corollary 5 (Worst-case probabilities over Gelbrich ambiguity sets).

If $\Xi\subseteq\mathbb{R}^{n}$ as a (not necessarily convex) Borel set,
$\mathcal{S}=\mathcal{M}_{2}$ and $\rho>0$, then we have

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}(\xi\in\Xi)=\left\\{\begin{array}[]{cl}\inf&y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\\\
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n},\;z\in\mathbb{R}_{+},\;Z\in\mathbb{S}_{+}^{n}\\\
&\begin{bmatrix}\gamma I-Y&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Y&\gamma\widehat{\mu}+y\\\
(\gamma\widehat{\mu}+y)^{\top}&z\end{bmatrix}\succeq
0,\;\begin{bmatrix}Y&y\\\\[2.15277pt] y^{\top}&y_{0}\end{bmatrix}\succeq
0\\\\[10.76385pt] &y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq
1\quad\forall\xi\in\Xi.\end{array}\right.$

###### Proof.

The probability $\mathbb{Q}(\xi\in\Xi)$ can be viewed as the expected value of
the indicator loss function $\ell=\mathds{1}_{\Xi}$ corresponding to the set
$\Xi$, which is defined through $\mathds{1}_{\Xi}(\xi)=1$ if $\xi\in\Xi$ and
$\mathds{1}_{\Xi}(\xi)=0$ if $\xi\not\in\Xi$. Thus, we can address worst-case
probability problems with the techniques of Theorem 6. Specifically, the
convex set $\mathcal{Y}$ associated with the indicator loss function can be
represented as

$\displaystyle\mathcal{Y}$
$\displaystyle=\left\\{y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n}:y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq\mathds{1}_{\Xi}(\xi)~{}\forall\xi\in\mathbb{R}^{n}\right\\}$
$\displaystyle=\left\\{y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n}:y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq
0~{}\forall\xi\in\mathbb{R}^{n},~{}y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq
1~{}\forall\xi\in\Xi\right\\}$
$\displaystyle=\left\\{y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n}:\begin{bmatrix}Y&y\\\\[2.15277pt]
y^{\top}&y_{0}\end{bmatrix}\succeq 0,\;y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq
1~{}\forall\xi\in\Xi\right\\}.$

The claim then follows by substituting this representation of $\mathcal{Y}$
into the problem of Theorem 6. ∎

###### Proof of Theorem 7.

By introducing an auxiliary variable $\tau$, the Gelbrich VaR can be recast as

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\text{-}\operatorname{VaR}_{\beta}(\ell(\xi))$
$\displaystyle=\inf_{\tau\in\mathbb{R}}\bigg{\\{}\tau:\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\text{-}\operatorname{VaR}_{\beta}(\ell(\xi))\leq\tau\bigg{\\}}$
$\displaystyle=\inf_{\tau\in\mathbb{R}}\bigg{\\{}\tau:\mathbb{Q}\text{-}\operatorname{VaR}_{\beta}(\ell(\xi))\leq\tau~{}\forall\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})\bigg{\\}}$
$\displaystyle=\inf_{\tau\in\mathbb{R}}\bigg{\\{}\tau:\mathbb{Q}\left(\ell(\xi)>\tau\right)\leq\beta~{}\forall\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})\bigg{\\}}$
$\displaystyle=\inf_{\tau\in\mathbb{R}}\bigg{\\{}\tau:\sup_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\left(\ell(\xi)>\tau\right)\leq\beta\bigg{\\}}$
$\displaystyle=\inf_{\tau\in\mathbb{R}}\bigg{\\{}\tau:\sup_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\left(\ell(\xi)\geq\tau\right)\leq\beta\bigg{\\}},$

where the last equality follows from a straightforward adaptation of the
arguments in [111, p. 175], which imply that the mapping
$\tau\mapsto\sup_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}(\ell(\xi)>\tau)$
is lower semi-continuous and non-increasing. Corollary 5 then allows us to
reformulate the worst-case probability on the last line as

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\left(\ell(\xi)\geq\tau\right)=\left\\{\begin{array}[]{cl}\inf&y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\gamma,z\in\mathbb{R}_{+},\;y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n},\;Z\in\mathbb{S}_{+}^{n}\\\\[4.30554pt]
&\begin{bmatrix}\gamma I-Y&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Y&\gamma\widehat{\mu}+y\\\
(\gamma\widehat{\mu}+y)^{\top}&z\end{bmatrix}\succeq
0,\;\begin{bmatrix}Y&y\\\\[2.15277pt] y^{\top}&y_{0}\end{bmatrix}\succeq
0\\\\[12.91663pt] &y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq
1\qquad\forall\xi\in\Xi_{\tau},\end{array}\right.$

where the uncertainty set
$\Xi_{\tau}=\\{\xi\in\mathbb{R}^{n}:\ell(\xi)\geq\tau\\}$ of the emerging
robust constraint depends on the decision variable $\tau$. Recalling the
polyhedrality of $\ell(\xi)$, this constraint is satisfied if and only if

$\displaystyle\left.\begin{array}[]{cl}\displaystyle\inf_{\xi\in\mathbb{R}^{n}}&\displaystyle
y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\\\\[4.30554pt]
\operatorname{s.t.}&\displaystyle\tau+w^{\top}\xi+w^{\top}\max\\{A\xi+a,B\xi+b\\}\leq
0\end{array}\right\\}\geq 1.$

By standard duality arguments akin to those used in [111, Theorem 2.3], this
inequality holds if and only if there exists a scalar $\eta\geq 0$ and a
vector $\zeta\in\mathbb{R}^{n}$, $\zeta\leq w$, such that

$\displaystyle\begin{bmatrix}Y&y\\\
y^{\top}&y_{0}\end{bmatrix}+\begin{bmatrix}0&\frac{\eta}{2}(w+(A-B)^{\top}\zeta+B^{\top}w)\\\
\frac{\eta}{2}(w+(A-B)^{\top}\zeta+B^{\top}w)^{\top}&\eta(\tau+(a-b)^{\top}\zeta+b^{\top}w)-1\end{bmatrix}\succeq
0.$

Substituting the emerging matrix inequality into our reformulation of the
worst-case probability
$\sup_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}(\ell(\xi)\geq\tau)$
then yields the following reformulation of the Gelbrich VaR problem.

$\displaystyle\begin{array}[]{cl}\inf&\tau\\\\[-4.30554pt]
\operatorname{s.t.}&\gamma,\eta,z\in\mathbb{R}_{+},~{}\tau,y_{0},v_{0}\in\mathbb{R},~{}v,y\in\mathbb{R}^{n},~{}\zeta\in\mathbb{R}_{+}^{k},~{}Y\in\mathbb{S}^{n},~{}Z\in\mathbb{S}_{+}^{n}\\\
&\zeta\leq
w,~{}y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\leq\beta\\\
&v=\frac{1}{2}\left(w+(A-B)^{\top}\zeta+B^{\top}w\right),~{}v_{0}=\tau+(a-b)^{\top}\zeta+b^{\top}w\\\
&\begin{bmatrix}\gamma I-Y&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Y&\gamma\widehat{\mu}+y\\\
(\gamma\widehat{\mu}+y)^{\top}&z\end{bmatrix}\succeq 0\\\\[12.91663pt]
&\begin{bmatrix}Y&y\\\\[2.15277pt] y^{\top}&y_{0}\end{bmatrix}\succeq
0,\;\begin{bmatrix}Y&y+\eta v\\\ y^{\top}+\eta v^{\top}&y_{0}+\eta
v_{0}-1\end{bmatrix}\succeq 0.\end{array}$ (47)

Note that problem (47) is still non-convex because of the bilinear terms in
the last constraint. To eliminate these bilinear terms, we first show that
$\eta>0$ at any feasible solution of (47). Assume to the contrary that there
exists a feasible solution with $\eta=0$. The last semidefinite constraint
then implies that $y_{0}\geq 1$. In addition, as $Y\succeq 0$ and thus $\gamma
I-Y\preceq\gamma I$, the first semidefinite constraint implies via a Schur
complement argument that
$\operatorname{Tr}\big{[}Z\big{]}\geq\gamma\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}$.
Similarly, the second and the fourth semidefinite constraints imply that
$z\geq\gamma\|\widehat{\mu}\|^{2}+1-y_{0}$. These insights collectively show
that

$y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\geq
1+\gamma\rho^{2}\geq 1,$

which contradicts the second constraint in (47) for any $\beta\in(0,1)$.
Hence, we may conclude that our assumption was false and that $\eta>0$.
Consequently, we can divide all constraints of problem (47) by $\eta$, and we
can apply the substitution
$(\gamma,\eta,z,y_{0},y,Y,Z)\leftarrow(\gamma/\eta,1/\eta,z/\eta,y_{0}/\eta,y/\eta,Y/\eta,Z/\eta)$.
In doing so, problem (47) reduces to the desired (tractable) semidefinite
program. ∎

###### Proof of Theorem 8.

We have

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\text{-}\operatorname{CVaR}_{\beta}(\ell(\xi))$
$\displaystyle=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\sup\limits_{\mathbb{Q}\in\mathcal{P}(\mu,\Sigma)}\mathbb{Q}\text{-}\operatorname{CVaR}_{\beta}(\ell(\xi))$
$\displaystyle=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\sup\limits_{\mathbb{Q}\in\mathcal{P}(\mu,\Sigma)}\mathbb{Q}\text{-}\operatorname{VaR}_{\beta}(\ell(\xi))$
$\displaystyle=\sup\limits_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}\text{-}\operatorname{VaR}_{\beta}(\ell(\xi)),$

where the first and the third equalities follow from the two-layer
decomposition (6a) of the Gelbrich ambiguity set, while the second equality
holds because of [111, Theorem 2.2]. ∎

###### Proof of Theorem 9.

The proof widely parallels that of Theorem 7 and can thus be kept short.
First, the Gelbrich VaR can be expressed as the optimal value of a
minimization problem over $\tau\in\mathbb{R}$ that involves the worst-case
probability
$\sup_{\mathbb{Q}\in\mathbb{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathbb{Q}(\ell(\xi)\geq\tau)$.
By Corollary 5, this worst-case probability can again be expressed as the
optimal value of a minimization problem with a robust constraint. Using strong
convex duality, this robust constraint can then be shown to be equivalent to a
matrix inequality. In summary, the Gelbrich VaR of a quadratic loss fucntion
thus equals

$\displaystyle\begin{array}[]{cl}\inf&\tau\\\\[-4.30554pt]
\operatorname{s.t.}&\gamma,z,\eta\in\mathbb{R}_{+},\;y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n},\;Z\in\mathbb{S}_{+}^{n},\;\tau\in\mathbb{R}\\\
&y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\leq\beta\\\
&\begin{bmatrix}Y&y\\\
y^{\top}&y_{0}\end{bmatrix}+\begin{bmatrix}\eta\Gamma(w)&\eta\Delta(w)\\\
\eta\Delta(w)^{\top}&-1+2\eta(\tau+\theta(w))\end{bmatrix}\succeq
0\\\\[12.91663pt] &\begin{bmatrix}\gamma
I-Y&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Y&\gamma\widehat{\mu}+y\\\
(\gamma\widehat{\mu}+y)^{\top}&z\end{bmatrix}\succeq
0,\;\begin{bmatrix}Y&y\\\\[2.15277pt] y^{\top}&y_{0}\end{bmatrix}\succeq
0.\end{array}$ (53)

An analogous reasoning as in the proof of Theorem 7 reveals that every
feasible solution of problem (53) satisfies $\eta>0$. The claim then follows
by dividing the constraints of problem (53) by $\eta$ and by applying a
similar variable substitution as in the proof of Theorem 7. ∎

###### Proof of Theorem 10.

The proof is analogous to that of Theorem 8 and thus omitted. ∎

The tractability of the generic worst-case expectation problem addressed in
Theorem 6 depends on the properties of the underlying loss function $\ell$.
The following corollary shows that this worst-case expectation problem is
equivalent to a tractable semidefinite programming whenever $\ell$ is defined
as the pointwise maximum of finitely many (possibly indefinite) quadratic
functions.

######  Corollary 6 (Piecewise quadratic loss).

Suppose that
$\ell(\xi)=\max_{j=1,\ldots,J}\\{\xi^{\top}Q_{j}\xi+2q_{j}^{\top}\xi+q^{0}_{j}\\}$
for some $Q_{j}\in\mathbb{S}^{n}$, $q_{j}\in\mathbb{R}^{n}$ and
$q_{j}^{0}\in\mathbb{R}$, $j=1,\ldots,J$. If $\mathcal{S}=\mathcal{M}_{2}$ and
$\rho>0$, then we have

$\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathds{E}_{\mathbb{Q}}[\ell(\xi)]=\left\\{\begin{array}[]{cl}\inf&y_{0}+\gamma\big{(}\rho^{2}-\|\widehat{\mu}\|^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+z+\operatorname{Tr}\big{[}Z\big{]}\\\
\operatorname{s.t.}&\gamma,z\in\mathbb{R}_{+},\;y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n},\;Z\in\mathbb{S}_{+}^{n}\\\
&\begin{bmatrix}\gamma I-Y&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\begin{bmatrix}\gamma I-Y&\gamma\widehat{\mu}+y\\\
(\gamma\widehat{\mu}+y)^{\top}&z\end{bmatrix}\succeq 0\\\\[12.91663pt]
&\begin{bmatrix}Y-Q_{j}&y-q_{j}\\\\[2.15277pt]
y^{\top}-q_{j}^{\top}&y_{0}-q^{0}_{j}\end{bmatrix}\succeq 0\quad\forall
j=1,\ldots,J.\end{array}\right.$

###### Proof.

By Theorem 6, the worst-case expectation problem at hand can be reformulated
as an equivalent minimization problem. The convex set $\mathcal{Y}$ appearing
in this reformulation satisfies

$\displaystyle\mathcal{Y}$
$\displaystyle=\left\\{y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n}:y_{0}+2y^{\top}\xi+\xi^{\top}Y\xi\geq\xi^{\top}Q_{j}\xi+2q_{j}^{\top}\xi+q^{0}_{j}~{}\forall\xi\in\mathbb{R}^{n},\,\forall
j=1,\ldots,J\right\\}$
$\displaystyle=\left\\{y_{0}\in\mathbb{R},\;y\in\mathbb{R}^{n},\;Y\in\mathbb{S}^{n}:\begin{bmatrix}Y-Q_{j}&y-q_{j}\\\\[2.15277pt]
y^{\top}-q_{j}^{\top}&y_{0}-q_{j}^{0}\end{bmatrix}\succeq 0~{}\forall
j=1,\ldots,J\right\\}.$

Thus, $\mathcal{Y}$ is semidefinite representable. The claim then follows by
substituting this representation of $\mathcal{Y}$ into the optimization
problem derived in Theorem 6. ∎

##  Appendix E Mean-Variance Risk Measures

Mean-variance risk measures appear in static portfolio selection [69, 36] and
in myopic reformulations of dynamic portfolio optimization [70], due to their
analytic tractability. For any probability distribution
$\mathbb{Q}\in\mathcal{M}$, the mean-variance risk measure with risk-aversion
coefficient $\beta\geq 0$ of any loss function $\ell\in\mathcal{L}_{0}$ is
defined as

$\mathcal{R}_{\mathbb{Q}}(\ell(\xi))=\mathds{E}_{\mathbb{Q}}[\ell(\xi)]+\beta\mathbb{V}\text{ar}_{\mathbb{Q}}(\ell(\xi)),$

where $\mathbb{V}\text{ar}_{\mathbb{Q}}(\ell(\xi))$ denotes the variance of
the loss $\ell(\xi)$ under the probability distribution $\mathbb{Q}$. Even
though the mean-variance risk measure gives rise to a law-invariant family of
translation invariant risk measures, it fails to be positive homogeneous, and
therefore Theorem 5 is not applicable. Nevertheless, the Gelbrich risk
evaluation problem can still be reformulated as a tractable second-order cone
program.

######  Theorem 12 (Gelbrich mean-variance risk of linear loss functions).

Suppose that $\\{\mathcal{R}_{\mathbb{Q}}\\}_{\mathbb{Q}\in\mathcal{M}}$ is a
family of mean-variance risk measures with coefficient $\beta>0$. If
$\widehat{\Sigma}\succ 0$, then the Gelbrich risk and the Wasserstein risk of
any portfolio loss function $\ell(\xi)=-w^{\top}\xi$ coincide and are equal to
the optimal value of the second-order cone program

$\begin{array}[]{cl}\inf&~{}\gamma\rho^{2}-\widehat{\mu}^{\top}w+\frac{1}{4}y+\beta
z\\\
\operatorname{s.t.}&~{}\gamma\in\mathbb{R}_{+},\;y\in\mathbb{R}_{+},\;z\in\mathbb{R}_{+}\\\\[8.61108pt]
&~{}\left\|\begin{pmatrix}2\widehat{\Sigma}^{\frac{1}{2}}w\\\ z+\beta
y-1\end{pmatrix}\right\|\leq z-\beta y+1,\;\left\|\begin{pmatrix}2w\\\
y-\gamma\end{pmatrix}\right\|\leq y+\gamma,\;\beta y\leq 1.\end{array}$ (54)

###### Proof of Theorem 12.

The equivalence of the Gelbrich risk and the Wasserstein risk for portfolio
loss functions follows from Proposition 1 and the observation that the mean-
variance risk measure depends on the distribution of the asset returns only
through its first and second moments. It remains to be shown that the Gelbrich
risk coincides with (54).

If $w=0$, then the Gelbrich risk evaluates to $0$, and problem (54) is solved
by $\gamma=y=z=0$. Thus, the claim is trivially satisfied. From now on we may
assume without loss of generality that $w\neq 0$. Using the decomposition
(6a), the Gelbrich risk can be recast as

$\displaystyle\mathcal{R}_{\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}(\ell)=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\sup\limits_{\mathbb{Q}\in\mathcal{C}(\mu,\Sigma)}\mathcal{R}_{\mathbb{Q}}(-w^{\top}\xi)=\sup\limits_{(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\left\\{-w^{\top}\mu+\beta
w^{\top}\Sigma w\right\\},$

where the second equality uses the definition of the mean-variance risk
measure. Note that the last optimization problem in the above expression
evaluates the support function of
$\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})$ at the point $(-w,\beta
ww^{\top})$, and by Proposition 13 we thus have

$\displaystyle\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;\mathcal{R}_{\mathbb{Q}}(-w^{\top}\xi)=\left\\{\begin{array}[]{cl}\inf&-\widehat{\mu}^{\top}w+\tau+\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\operatorname{Tr}\big{[}Z\big{]}\\\\[4.30554pt]
\operatorname{s.t.}&\gamma\in\mathbb{R}_{+},\;\tau\in\mathbb{R}_{+},\;Z\in\mathbb{S}^{n}_{+}\\\\[4.30554pt]
&\begin{bmatrix}\gamma I-\beta
ww^{\top}&\gamma\widehat{\Sigma}^{\frac{1}{2}}\\\
\gamma\widehat{\Sigma}^{\frac{1}{2}}&Z\end{bmatrix}\succeq
0,\;\left\|\begin{pmatrix}-w\\\
\tau-\gamma\end{pmatrix}\right\|\leq\tau+\gamma.\end{array}\right.$

Fix now any feasible solution $(\gamma,\tau,Z)$ of the resulting semidefinite
program. The first matrix inequality implies that $\gamma I\succeq\beta
ww^{\top}$. As $\beta>0$ and $w\neq 0$, this is only possible if $\gamma>0$,
which in turn implies that $\gamma\widehat{\Sigma}^{\frac{1}{2}}\succ 0$. By
[7, Corollary 8.2.2], the first matrix inequality in the above semidefinite
program therefore also implies that $\gamma I-\beta ww^{\top}\succ 0$, which
is equivalent to $\gamma>\beta\|w\|^{2}$. It is easy to verify that, at
optimality, $\tau$ coincides with $\|w\|^{2}/(4\gamma)$ and $Z$ coincides with
its Schur complement $\gamma^{2}(\gamma I-\beta
ww^{\top})^{-1}\widehat{\Sigma}$, which is well-defined because $\gamma
I-\beta ww^{\top}\succ 0$. Thus, $\tau$ and $Z$ can be eliminated together
with their constraints to obtain

$\displaystyle\phantom{=}\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\;\mathcal{R}_{\mathbb{Q}}(-w^{\top}\xi)$
$\displaystyle=\inf_{\gamma>\beta\|w\|^{2}}-\widehat{\mu}^{\top}w+\frac{\|w\|^{2}}{4\gamma}+\gamma\big{(}\rho^{2}-\operatorname{Tr}\big{[}\widehat{\Sigma}\big{]}\big{)}+\gamma^{2}\operatorname{Tr}\big{[}(\gamma
I-\beta ww^{\top})^{-1}\widehat{\Sigma}\big{]}$
$\displaystyle=\inf\limits_{\gamma>\beta\|w\|^{2}}\gamma\rho^{2}-\widehat{\mu}^{\top}w+\frac{\|w\|^{2}}{4\gamma}+\beta(1-\beta\gamma^{-1}\|w\|^{2})^{-1}w^{\top}\widehat{\Sigma}w,$

where the second equality follows from the Sherman-Morrison formula [7,
Corollary 2.8.8]. Introducing an auxiliary variable $y\geq 0$ and rewriting
the constraint in the last expression as $\|w\|^{2}/\gamma\leq
y\leq\beta^{-1}$ yields

$\sup\limits_{\mathbb{Q}\in\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})}\mathcal{R}_{\mathbb{Q}}\left(-w^{\top}\xi\right)=\left\\{\begin{array}[]{cl}\inf&\gamma\rho^{2}-\widehat{\mu}^{\top}w+\frac{y}{4}+\beta(1-\beta
y)^{-1}w^{\top}\widehat{\Sigma}w\\\ \operatorname{s.t.}&\|w\|^{2}/\gamma\leq
y\leq\beta^{-1},\;\gamma>0.\end{array}\right.$

As $\gamma\geq 0$, $y\geq 0$ and $\beta y\leq 1$, we may use [64, Equation
(8)] to reformulate the hyperbolic constraint and the quadratic-over-linear
term in the objective function in terms of second-order cone constraints, that
is,

$\displaystyle\|w\|^{2}\leq\gamma y~{}$
$\displaystyle\iff~{}\left\|\begin{pmatrix}2w\\\
y-\gamma\end{pmatrix}\right\|\leq
y+\gamma\quad\text{and}\quad\frac{w^{\top}\widehat{\Sigma}w}{1-\beta y}\leq
z~{}$
$\displaystyle\iff~{}\left\|\begin{pmatrix}2\widehat{\Sigma}^{\frac{1}{2}}w\\\
z+\beta y-1\end{pmatrix}\right\|\leq z-\beta y+1,$

where $z\geq 0$ is an auxiliary epigraphical variable. Thus, the claim
follows. ∎

In analogy to Proposition 6, we can determine the first and second moments of
the worst-case probability distributions that maximize the Gelbrich mean-
variance risk of a fixed linear loss function.

######  Proposition 15 (Worst-case moments).

Suppose that $\\{\mathcal{R}_{\mathbb{Q}}\\}_{\mathbb{Q}\in\mathcal{M}}$ is a
family of mean-variance risk measures with $\beta>0$. If
$\widehat{\Sigma}\succ 0$, $w\neq 0$ and $\rho>0$, then any extremal
distribution $\mathbb{Q}^{\star}$ that attains the Gelbrich risk of the linear
loss function $\ell(\xi)=-w^{\top}\xi$ has the same mean
$\mu^{\star}\in\mathbb{R}^{n}$ and covariance matrix
$\Sigma^{\star}\in\mathbb{S}_{+}^{n}$, where

$\displaystyle\mu^{\star}=\widehat{\mu}-\frac{w}{2\gamma^{\star}},\quad\Sigma^{\star}=\left(I-\frac{\beta
ww^{\top}}{\gamma^{\star}}\right)^{-1}\widehat{\Sigma}\left(I-\frac{\beta
ww^{\top}}{\gamma^{\star}}\right)^{-1}$

and $\gamma^{\star}>\beta\|w\|^{2}$ is the unique solution of the nonlinear
algebraic equation

$\displaystyle\frac{\|w\|^{2}}{4\gamma^{2}}+\operatorname{Tr}\big{[}\widehat{\Sigma}\left(I-\gamma(\gamma
I-\beta ww^{\top})^{-1}\right)^{2}\big{]}=\rho^{2}.$

We emphasize again that there may be multiple extremal distributions in the
Gelbrich ambiguity set $\mathcal{G}_{\rho}(\widehat{\mu},\widehat{\Sigma})$
that share the unique extremal mean $\mu^{\star}$ and covariance matrix
$\Sigma^{\star}$ identified in Proposition 15.

###### Proof of Proposition 15.

From the proof of Theorem 12 we know that the first and second moments of any
extremal distribution $\mathbb{Q}^{\star}$ are maximizers of the support
function evaluation problem
$\sup\\{q^{\top}\mu+\operatorname{Tr}\big{[}Q\Sigma\big{]}:(\mu,\Sigma)\in\mathcal{U}_{\rho}(\widehat{\mu},\widehat{\Sigma})\\}$
with $q=-w$ and $Q=\beta ww^{\top}$. The claim thus follows from Lemma 2,
which applies because $w\neq 0$. ∎

Distributionally robust mean-variance portfolio optimization problems with
2-Wasserstein ball ambiguity sets were also studied by [12]. Instead of
minimizing a worst-case mean-variance risk measure, however, they minimize the
worst-case variance of the portfolio return subject to a lower bound on the
worst-case mean, which results in a more conservative model because the
extremal distriburtions in the objective function and in the constraints may
differ. The additional conservatism enhances analytical tractability.