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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!

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1 Answer 1

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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.)

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