Let $\mathcal{P}$ be the set of real-valued and strictly stationary processes with expectation zero and finite variance, i.e.:
\begin{equation}
 \mathcal{P}:=\left\{ X = (X\_t)\_{t \in \mathbb{Z}} \, : \, X \hbox{ is strictly stationary, } \mathbb{E} X\_t = 0 \hbox{ and } \mathbb{E}[X\_t^2]< \infty, \, \forall\, t \in \mathbb{Z} \right\}
\end{equation}
*Remark:* for any stochastic process $X$, we consider $Q$ the Law of a stochastic process according [this](https://en.wikipedia.org/wiki/Law_(stochastic_processes)). We denote $X \sim Q$.


I'm trying to show whether or not $\mathcal{P}$ is closed according to the Mallows metric:


Let $X = (X\_t)\_{t \in \mathbb{Z}} \sim P$ and $Y = (Y\_t)\_{t \in \mathbb{Z}}\sim Q$ be two stochastic processes. In order to define the Mallows metric, for all $m\in \mathbb{N}$, let $\mathcal{M}\_m$ be the random vectors $(\tilde{X},\tilde{Y})$ having marginals $P\circ\pi\_{1,...,m}^{-1}$ and $Q\circ\pi\_{1,...,m}^{-1}$, where $\pi\_{1,...,m}( (X\_t)\_{t \in \mathbb{Z}} )= (X\_{1},..., X\_{m})$ . So:
$$d( (X\_t)\_{t \in \mathbb{Z}},(Y\_t)\_{t \in \mathbb{Z}})= \sum\_{m=1}^\infty d^{(m)}(P\circ\pi\_{1,...,m}^{-1}, Q\circ\pi\_{1,...,m}^{-1})2^{-m}$$
where
$$d^{(m)}(P\circ\pi\_{1,...,m}^{-1}, Q\circ\pi\_{1,...,m}^{-1}) = \inf\_{(\tilde{X},\tilde{Y})\in \mathcal{M}\_m}{(E||\tilde{X}-\tilde{Y}||^2)^{\tfrac{1}{2}}}.$$


Some hint?



Show that the set of strictly stationary, mean zero and finite variance stochastic processes is closed (or not)
$\newcommand{\Z}{\mathbb Z}\newcommand{\PP}{\mathcal D}\newcommand{\R}{\mathbb R}$Your function $d$ is not a metric, for two reasons: (i) there may be many processes $(X\_t)\_{t\in\Z}$ with the same distribution $P$ and (ii) your function $d$ does not take into account the values of $X\_t$ for negative $t\in\Z$. So, your $d$ is, not a metric, but a [pseudometric](https://en.wikipedia.org/wiki/Pseudometric_space), which does not allow one to identify limits uniquely.


We can fix these deficiencies as follows: Let $\PP$ denote the set of the distributions of the processes in $\mathcal P$.


Given $P$ and $Q$ in $\PP$, for any natural $m$ let
\begin{equation}
 P\_m:=P\circ\pi\_{-m,\dots,m}^{-1},\quad Q\_m:=Q\circ\pi\_{-m,\dots,m}^{-1}, 
\end{equation}
where $\pi\_{r,\dots,s}((x\_t)\_{t\in\Z}):=(x\_r,\dots,x\_s)$ for any given integers $r,s$ such that $r\le s$.
Let
\begin{equation}
 d(P,Q):=\sum\_{m=1}^\infty d^{(m)}(P\_m,Q\_m)2^{-m}, 
\end{equation}  

where $d^{(m)}$ is the [Wasserstein metric](https://en.wikipedia.org/wiki/Pseudometric_space) of order $2$.


We want then to show that $\PP$ is closed with respect to the metric $d$.


Suppose now that we have a sequence $(P^{(n)})$ in $\PP$ such that $d(P^{(n)},Q)\to0$ (as $n\to\infty$) for some probability measure $Q$ (on the [cylindrical $\sigma$-algebra](https://en.wikipedia.org/wiki/Cylindrical_%CF%83-algebra)) over $\R^\Z$. Then for each natural $m$ we have $d^{(m)}(P^{(n)}\_m,Q\_m)\to0$. So, by the well-known [characterization of the convergence in the Wasserstein metric](https://math.stackexchange.com/questions/3042947/characterization-of-wasserstein-convergence), $P^{(n)}\_m\to Q\_m$ weakly,
$\int\_{\R^{\Z\_m}} x\_t^2\,Q\_m(dx)=\lim\_n\int\_{\R^{\Z\_m}} x\_t^2\,P^{(n)}\_m(dx)<\infty$, and
$\int\_{\R^{\Z\_m}} x\_t\,Q\_m(dx)=\lim\_n\int\_{\R^{\Z\_m}} x\_t\,P^{(n)}\_m(dx)=\lim\_n0=0$ for $t\in\Z\_m:=\{-m,\dots,m\}$.


So, $\int\_{\R^\Z} x\_t\,Q(dx)=0$ and $\int\_{\R^\Z} x\_t^2\,Q(dx)<\infty$ for all $t\in\Z$, and
$P^{(n)}\_{r,s}\to Q\_{r,s}$ weakly for any given integers $r,s$ such that $r\le s$, where $P^{(n)}\_{r,s}:=P^{(n)}\circ\pi\_{r,\dots,s}^{-1}$ and $Q\_{r,s}:=Q\circ\pi\_{r,\dots,s}^{-1}$.


By the stationarity, $P^{(n)}\_{r+1,s+1}=P^{(n)}\_{r,s}$ for all suitable $r,s,n$. Letting now $n\to\infty$, we conclude that $Q\_{r+1,s+1}=Q\_{r,s}$, so that $Q$ is the distribution of a stationary process. Also, as we saw, $\int\_{\R^\Z} x\_t\,Q(dx)=0$ and $\int\_{\R^\Z} x\_t^2\,Q(dx)<\infty$ for all $t\in\Z$. So, $Q\in\PP$.


We conclude that $\PP$ is closed, as desired.


