Q:

Linear transform of a strictly stationary time series

First, let me clarify what I mean by a strictly stationary time series.  Let $(X_t)_{t\in \mathbb{Z}}$ be a sequence of random variables on some probability space.  If it holds that
$$(X_t, X_{t+1},\ldots,X_{t+h}) \stackrel{d}{=} (X_s, X_{s+1},\ldots,X_{s+h})$$
for every $s,t \in \mathbb{Z}$ and every $h \in \mathbb{Z}_{\geq 0}$.
Next, I will explain what I mean by linear transform.  Given a sequence of reals $(\psi_j)_{j\in \mathbb{Z}}$ such that $\sum_j\lvert\psi_j\rvert < \infty$,  linear transform of $(X_t)$, which we denote by $(Y_t)$, is defined as
$$Y_t = \sum_j\psi_jX_{t-j}$$
Now suppose that the sum above is well-defined, i.e. $\sum_j\psi_jX_{t-j}$ is finite almost surely. ( $E\lvert X_t\rvert < \infty$ would be sufficient for this but we don't assume that.) 
How do I then show that $(Y_t)$ is also strictly stationary?  Intuitively speaking, 
$$\sum_j\psi_jX_{t-j} \stackrel{d}{=} \sum_j\psi_jX_{s-j}$$
since $(X_t)$ is strictly stationary.  But shifting a finite sequence is not quite the same as ``shifting an infinite sequence".  How do I make this rigorous? 

A:

Fix some $s,t \in \Bbb Z$. Note that since $(X_t, X_{t+1},\ldots,X_{t+h}) \stackrel{d}{=} (X_s, X_{s+1},\ldots,X_{s+h})$ for all $h$, we have that $$(X_{t+k})_{k \in \Bbb Z} \stackrel{d}{=} (X_{s+k})_{k \in \Bbb Z}$$ since both processes have the same finite-dimensional distributions.  If you are not familiar with this fact from elementary measure theory, see for example Theorem 23 here (there $\Xi$ denotes the space of functions endowed with product $\sigma$-algebra) or alternatively Proposition 3.1 here. 
Now define the map $\psi: \Bbb R^{\Bbb Z} \to \Bbb R^{\Bbb Z}$ by sending $(x_j)_{j \in \Bbb Z} \mapsto (\sum_j \psi_j x_{k-j})_{k \in \Bbb Z}$ and note that it is measurable on the product $\sigma$-algebra (since the individual components are).  Fix some $s,t \in \Bbb Z$, and since $(X_{t+k})_{k \in \Bbb Z} \stackrel{d}{=} (X_{s+k})_{k \in \Bbb Z}$ we easily get that $$(Y_{t+k})_{k\in\Bbb Z}=\psi\big((X_{t+k})_{k \in \Bbb Z}\big) \stackrel{d}{=} \psi\big((X_{s+k})_{k \in \Bbb Z}\big)=(Y_{s+k})_{k\in\Bbb Z}$$ which precisely means stationarity of $Y$.

