I want to understand an approximation of a compound Poisson distribution in this [paper](https://link.springer.com/content/pdf/10.1023/A:1022616601841.pdf?pdf=button).


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](https://mathoverflow.net/questions/428657/show-that-the-set-of-strictly-stationary-mean-zero-and-finite-variance-stochast), $\mathcal{P}$ 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](https://en.wikipedia.org/wiki/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](https://link.springer.com/content/pdf/10.1023/A:1022616601841.pdf?pdf=button) 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](https://link.springer.com/content/pdf/10.1023/A:1022616601841.pdf?pdf=button) ):
\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:


1. $X\_{t\_1,...,t\_m}^{(n)}\implies Y\_{t\_1,...,t\_m}\, (n \to \infty)$ 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.
2. $E[|X^{(n)}\_{t}|^2] \to E[|Y\_{t}|^2], (n \to \infty)$ 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](https://mathoverflow.net/questions/436873/approximation-of-a-random-sum-of-random-variables-infinitely-divisible-distribu/), 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](https://link.springer.com/content/pdf/10.1023/A:1022616601841.pdf?pdf=button) .


Any clarification?



Understanding the approximation of a random sum of random processes
The paper seems to be written rather carelessly. In particular, it is indeed unhelpful to denote a random object and its realizations by the same symbol, leaving the job of figuring out which is which here or there to the reader.


Further, it is clear that any sequence (say, the zero sequence) can be a realization of a random sequence $\xi$ (if e.g. $\xi$ is the sequence of iid normal random variables).


So, the result should be true, not for all realizations of $\xi$, but for **almost** all of them (excluding a set of realizations of probability $0$).


Indeed, the central point in the proof seems to be the use of the ergodic theorem in the middle of page 466 of the paper, where the convergence must of course be in the almost sure sense (but this is not specified on the paper).


