A linear process $(X\_t)\_{t \in \mathbb{Z}}$ is usually written as a moving-average process with infinity order:
\begin{equation}\label{linear\_process}\tag{Eq. 1.1}
 X\_{t} = \sum\_{j =0 }^\infty \psi\_{j} \varepsilon\_{t-j}, \forall t \in \mathbb{Z}.
\end{equation}
where $\varepsilon\_{t}$ is a i.i.d. white noise ($E(\varepsilon\_{t})=0,\,\, E |\varepsilon\_{t}|^2< \infty$) and $\sum\_{j =0 }^\infty \psi\_{j}^2 < \infty$.


According to this [paper](https://www.pnas.org/doi/pdf/10.1073/pnas.93.22.12128), page 12130, the author says:


Mallows (12) argues that a linear process such as in (\ref{linear\_process}) is close to a Gaussian process if $\max\_{j\geq 0}|\psi\_j|$ is small.


I would like to know if you have any relatively simple examples for this statement. I tried to think of a simple example, but I couldn't. I don't want to go to the Mallows paper before I go through here.



Linear process close to a Gaussian process
$\newcommand{\R}{\mathbb R}\newcommand{\Z}{\mathbb Z}\newcommand{\ep}{\varepsilon}\newcommand{\de}{\delta}$Let $\psi\_j:=0$ for $j=-1,-2,\dots$. Then
\begin{equation\*}
 X\_t=\sum\_{j\in\Z}X\_{t,j}
\end{equation\*}
for $t\in\Z$, where
\begin{equation\*}
 X\_{t,j}:=\psi\_{t-j}\ep\_j. 
\end{equation\*}
Let
\begin{equation\*}
 B:=\sqrt{\sum\_{j\in\Z} \psi\_i^2},\quad m:=\max\_{j\ge0}|\psi\_j|=\max\_{j\in\Z}|\psi\_j|. 
\end{equation\*}
Suppose that $B>0$ and $m$ vary in any manner such that
\begin{equation\*}
 m/B\to0. \tag{1}\label{1}
\end{equation\*}
Let us show that then $X\_t/B$ converges in distribution to a standard normal random variable, for each $t\in\Z$.




---


For each real $\de>0$,
\begin{equation\*}
\begin{aligned}
 L&:=\frac1{B^2}\sum\_{j\in\Z}EX\_{t,j}^2\,1(|X\_{t,j}|\ge\de B) \\ 
 &=\frac1{B^2}\sum\_{j\in\Z}E(\psi\_{t-j}\ep\_j)^2\,1(|\psi\_{t-j}\ep\_j|\ge\de B) \\ 
 &\le\frac1{B^2}\sum\_{j\in\Z}\psi\_{t-j}^2E\ep\_j^2\,1(|\ep\_j|\ge\de B/m) \\ 
 &=\frac1{B^2}\sum\_{j\in\Z}\psi\_{t-j}^2E\ep\_0^2\,1(|\ep\_0|\ge\de B/m) \\ 
 &=E\ep\_0^2\,1(|\ep\_0|\ge\de B/m)\to0. 
\end{aligned}
\end{equation\*}
Hence,
\begin{equation\*}
 \frac1{B^2}\sum\_{j\in\Z}EX\_{t,j}^2\,1(|X\_{t,j}|<\de B)=1-L\to1,
\end{equation\*}
\begin{equation\*}
\begin{aligned}
 &\frac1{B^2}\sum\_{j\in\Z}(EX\_{t,j}\,1(|X\_{t,j}|<\de B))^2 \\ 
 &=\frac1{B^2}\sum\_{j\in\Z}(EX\_{t,j}\,1(|X\_{t,j}|\ge\de B))^2
 \le L\to0,
\end{aligned}
\end{equation\*}
\begin{equation\*}
\begin{aligned}
 &\Big|\frac1B\sum\_{j\in\Z}EX\_{t,j}\,1(|X\_{t,j}|<\de B)\Big| \\ 
 &=\Big|\frac1B\sum\_{j\in\Z}EX\_{t,j}\,1(|X\_{t,j}|\ge\de B)\Big| \\ 
 &\le\frac1B\sum\_{j\in\Z}E|X\_{t,j}|\,1(|X\_{t,j}|\ge\de B)\
 \le \frac L\de\to0,
\end{aligned}
\end{equation\*}
\begin{equation\*}
 \sum\_{j\in\Z}P(|X\_{t,j}|\ge\de B)\le L\to0. 
\end{equation\*}
So, by [Theorem 18 in Chapter IV](https://link.springer.com/book/10.1007/978-3-642-65809-9), $X\_t/B$ converges in distribution to a standard normal random variable, for each $t\in\Z$.


Thus, under condition \eqref{1}, all the one-dimensional distributions of the process $(X\_t)$ are asymptotically normal.




---


Similarly considered are all the finite-dimensional distributions of the process $(X\_t)$ -- that is, all the joint distributions of $(X\_{t\_1},\dots,X\_{t\_p})$ for integers $t\_1<\cdots<t\_p$. This is done by writing
\begin{equation\*}
 \sum\_{i=1}^p c\_i X\_{t\_i}=\sum\_{j\in\Z}Y\_j
\end{equation\*}
for any real $c\_1,\dots,c\_p$, where
\begin{equation\*}
 Y\_j:=\phi\_j\ep\_j,\quad\phi\_j:=\sum\_{i=1}^p c\_i \psi\_{t\_i-j}, 
\end{equation\*}
so that $\sum\_{j\in\Z}\phi\_j^2<\infty$ and $\max\_{j\in\Z}|\phi\_j|\le m\sum\_{i=1}^p |c\_i|$.


