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. 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 $\pi_{1,...,m}( (X_t)_{t \in \mathbb{Z}} )= (X_{t_1},..., X_{t_m})$ . 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 $$d^{(m)}(P\circ\pi_{1,...,m}^{-1}, Q\circ\pi_{1,...,m}^{-1}) = \inf_{(\tilde{X},\tilde{Y})\in \mathcal{M}_m}{(E||X-Y||^2)^{\tfrac{1}{2}}}.$$

Some hint?