I want to understand an approximation of a compound Poisson distribution in this paper.

First, let's set the environment. Consider $\mathcal{P}$ the class of distributions of real-valued and strictly stationary processes with expectation zero and finite variance. According to this topic, $\mathcal{P}$ is closed with respect to the Mallows metric $d$ (see the topic for a formal definition of $d$). Abusing the notation, we write $X \in \mathcal{P}$ to say that the law of $X$ is in $\mathcal{P}$.

Given $N \sim \hbox{Poisson}(\lambda)$. Let $\xi = (\xi_t)_{t \in \mathbb Z}$ be an ergodic process in $\mathcal{P}$ and $\xi_1,\xi_2, \xi_3,... \overset{iid}{\sim} \xi$ sequence of stochastic process independent of $N$. Define the compound Poisson stochastic process $Y = (Y_t)_{t \in \mathbb Z}$ in $\mathcal{P}$: \begin{equation} Y_t = \sum_{j=1}^N \xi_{t;j}, \quad N \sim \hbox{Poisson}(\lambda) \end{equation} (warning: $Y$ is not the classic Compound Poisson process)

This is a particular case with $(W_t)_{t \in \mathbb{Z}}\equiv 0$, $k=1$ of the Step 2 of the proof of item (ii) of Theorem 1 from the same paper cited above (The statement of the theorem is on page 454 and its proof on page 465). According to the same paper, we can approximate $Y$ by the following sequence of linear processes $(X^{(n)} , n \geq 1)$ (See equation 5.11 on the paper ): \begin{equation} X^{(n)}_t = \sum_{j =1 }^n \bar{\xi}_j U_{t - j ;n} \end{equation} where $(\bar{\xi}_j)_{j\in \mathbb Z}$ is a fixed realization of $\xi = (\xi_t)_{t \in \mathbb Z}$ and $(U_{t;n})_{t \in \mathbb Z} \overset{iid}{\sim} \hbox{Bernoulli}(\lambda/n)$ independent of $\xi$. The approximation or the convergence is with respect to the Mallows metric $d$. The following characterization is useful: $d(X^{(n)},Y) \to 0,\,(n \to \infty)$ is equivalent to:

$X_{t_1,...,t_m}^{(n)}\implies Y_{t_1,...,t_m}\, (n \to \infty)$ for all $t_1,...,t_m \in \mathbb{Z}$ and all $m \in \mathbb{N}$. This is a convergence in distributions, and the method of characteristic functions can be used.

$E[|X^{(n)}_{t}|^2] \to E[|Y_{t}|^2], (n \to \infty)$ for any $t$.

For me, the proof is strange and confusing because $(\bar{\xi}_j)_{j\in \mathbb Z}$ sometimes is fixed realization, and sometimes it is treated as random. This causes conflict with items 1 and 2 above. For example:

To demonstrate the first item, the convergence of the characteristic functions $\varphi^{(n)}(s) \to \varphi_Y(s)$ has to be point-wise, and the paper delivers a convergence in probability. Still in this case, I tried to adapt this answer, but I don't have $(\bar{\xi}_j)_{j\in \mathbb Z}$ iid. So it seems that necessarily I have to first assume $(\bar{\xi}_j)_{j\in \mathbb Z}$ as a fixed realization and then treat it as random. Very strange!

For the second item, the convergence of expectations would have to be a convergence of real numbers, but the paper once again delivers a convergence in probability, first treating $(\bar{\xi}_j)_{j\in \mathbb Z}$ as fixed and then as random. See equation (5.13) on the paper .

Any clarification?