$f(\mathbf{y})-(f^{\star}-\varepsilon)-\mathscr{L}_{{\mathbf{S}}^{\star}}(P(\mathbf{y},\mathbf{x}))>0,$

for all $\mathbf{y}\in\mathcal{Y}$. As Assumption 4(i) holds, by Putinar’s
Positivstellensatz [44] (see also Theorem 2.5), there exists
$k_{2}^{(\epsilon)}\in\mathbb{N}$, such that for all $k\geq
k_{2}^{(\epsilon)}$, $(f^{\star}-\varepsilon,{\mathbf{S}}^{\star})$ is
feasible to (36) and hence $f_{k}^{\mbox{\tiny primal}}\geq
f^{\star}-\varepsilon$.

As $\varepsilon>0$ is arbitrary, by the weak duality, we have
$\lim_{k\to\infty}f^{\mbox{\tiny primal}}_{k}=\lim_{k\to\infty}f^{\mbox{\tiny
dual}}_{k}=f^{\star}$.

(ii). For each ${\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}$, define

$N({\boldsymbol{\alpha}})\coloneqq\sqrt{\binom{\ell+\left\lceil\frac{|{\boldsymbol{\alpha}}|}{2}\right\rceil}{\ell}}\sum_{i=1}^{\left\lceil\frac{|{\boldsymbol{\alpha}}|}{2}\right\rceil}b^{2i}.$

By Assumption 4(i) and Remark 4.15,
$|s^{(k,\star)}_{{\boldsymbol{\alpha}}}|\leq N({\boldsymbol{\alpha}})$ for all
$k$ and ${\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}_{2k}$. Hence, there always
exists a convergent subsequence of
$(\mathbf{s}^{(k,\star)}_{\mathbf{e}})_{k}$. Without loss of generality, we
assume that the whole sequence $(\mathbf{s}^{(k,\star)}_{\mathbf{e}})_{k}$
converges. Complete each $\mathbf{s}^{(k,\star)}$ with zeros to make it an
infinite vector. Then,

$\left\\{(s^{(k,\star)}_{{\boldsymbol{\alpha}}})_{{\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}}\right\\}_{k}\subseteq\prod_{{\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}}\left[-N({\boldsymbol{\alpha}}),N({\boldsymbol{\alpha}})\right].$

By Tychonoff’s theorem, the product space
$\prod_{{\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}}\left[-N({\boldsymbol{\alpha}}),N({\boldsymbol{\alpha}})\right]$
is compact in the product topology. Therefore, there exists a subsequence
$(\mathbf{s}^{(k_{i},\star)})_{i}$ of $(\mathbf{s}^{(k,\star)})_{k}$ and
$\mathbf{s}^{\star}=(s^{\star}_{{\boldsymbol{\alpha}}})_{{\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}}$
such that
$\lim_{i\to\infty}s^{(k_{i},\star)}_{{\boldsymbol{\alpha}}}=s^{\star}_{{\boldsymbol{\alpha}}}$
for all ${\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}$. By the pointwise
convergence, we have (a) $\mathbf{s}^{\star}\in\mathcal{M}_{k}(\Theta)$ for
all $k$; (b) $\mathscr{H}_{\mathbf{s}^{\star}}(1)=1$; (c)
$\mathscr{H}_{\mathbf{s}^{\star}}(P(\mathbf{y},\mathbf{x}))\succeq 0$ for all
$x\in\mathcal{X}$. By (a), (b), Putinar’s Positivstellensatz and Haviland’s
theorem, $\mathbf{s}^{\star}$ has a representing probability measure $\mu$
supported on $\mathcal{Y}$, i.e.,
$s^{\star}_{{\boldsymbol{\alpha}}}=\int_{\mathcal{Y}}\mathbf{y}^{{\boldsymbol{\alpha}}}\mathrm{d}\mu(\mathbf{y})$
for all ${\boldsymbol{\alpha}}\in\mathbb{N}^{\ell}$. By Lemma 5.1,
$P(\mathbf{s}^{\star}_{\mathbf{e}},\mathbf{x})\succeq\mathscr{H}_{\mathbf{s}^{\star}}(P(\mathbf{y},\mathbf{x}))\succeq
0$ and
$\theta_{i}(\mathbf{s}^{\star}_{\mathbf{e}})\geq\mathscr{H}_{\mathbf{s}^{\star}}(\theta_{i})\geq
0$, $i=1,\ldots,s$. Hence $\mathbf{s}^{\star}_{\mathbf{e}}$ is feasible to
(RPMIO). Moreover, as $f(\mathbf{y})$ is convex in $\mathbf{y}$, by (i) and
the pointwise convergence,

$f^{\star}=\mathscr{H}_{\mathbf{s}^{\star}}(f)\geq
f(\mathbf{s}^{\star}_{\mathbf{e}}),$

which indicates that $\mathbf{s}^{\star}_{\mathbf{e}}$ is a minimizer of
(RPMIO).