# Approximation of a stationary process by a sequence of ergodic and stationary sequence of stochastic processes

Let $$X = [X_t : t \in \mathbb{Z}] \sim P$$ and $$Y = [Y_t : t \in \mathbb{Z}]\sim Q$$ be two stochastic processes. Let's define the Mallows metric. Let $$\mathcal{M}_m$$ be the random vectors $$(X,Y)$$ having marginals $$P\circ\pi_{1,...,m}^{-1}$$ and $$Q\circ\pi_{1,...,m}^{-1}$$. So: $$d(X,Y)= \sum_{m=1}^\infty d^{(m)}(P\circ\pi_{1,...,m}^{-1}, Q\circ\pi_{1,...,m}^{-1})$$ where $$d^{(m)}(P\circ\pi_{1,...,m}^{-1}, Q\circ\pi_{1,...,m}^{-1}) = \inf_{(X,Y)\in \mathcal{M}_m}{(E||X-Y||^2)^{\tfrac{1}{2}}}.$$ The following characterization is useful: $$d(X_n,X) \to 0,\,(n \to \infty)$$ is equivalent to

1. $$X_{t_1,...,t_m;n}\implies X_{t_1,...,t_m}\, (n \to \infty)$$ for all $$t_1,...,t_m \in \mathbb{Z}, m \in \mathbb{N}$$
2. $$E[X_{0,n}^2] \to E[X_{0}^2], (n \to \infty)$$.

i.e., all finite dimensional distributions at $$t_1,...,t_m$$ converge weakly and the variance of the one-dimensional marginal converges.

Now I want to show the following:

given $$X = [X_t : t \in \mathbb{Z}] \sim P$$ be a stationary process with $$E[X_t]=0$$. Then there exists a sequence of stationary and ergodic processes $$(X^{n})_{n \in \mathbb{N}}$$ with $$E[X_t^n]=0$$ for all $$n$$ such that

$$d(X_n,X) \to 0,\,(n \to \infty)$$

Help!

Let's assume that $$P$$ has finitely many ergodic components so that $$P = \sum_{i=1}^m p_i P_i$$ with $$P_i$$ ergodic and $$\sum p_i = 1$$. Take $$X_i$$ to be independent stochastic processes with $$X_i \sim P_i$$ and let $$z$$ be a stationary irreducible continuous-time Markov process on $$\{1,\ldots,m\}$$ with invariant measure $$p$$ (and independent of the $$X_i$$'s). The sequence of processes $$X^n(t) = X_{z(t/n)}(t)$$ then has the desired property. For arbitrary $$P$$ I would suspect that a similar construction still works by first approximating $$P$$ by a law with finitely many ergodic components. (Maybe you need a little bit of extra tightness for that.)