# Showing a moving average is strictly stationary if underlying sequence is strictly stationary.

Just as the title suggests, this is my problem:

Let $Z_t$ be a strictly stationary sequence. Define $X_t = Z_t + \theta Z_{t-1}$. Show that this sequence is also strictly stationary.

Here's my problem. My definition of strictly stationary is that we have the distribution of $(Z_t,Z_{t+1},\dots,Z_{t+h})$ is independent of $t$ for all $t \in \mathbb{Z}$ and all $h \in \mathbb{N}$.

But how I see it we have $(X_t,X_{t+1},\dots,X_{t+h}) = (Z_t + \theta Z_{t-1},\dots,Z_{t+h} + \theta Z_{t+h-1})$ which would be independent of $t-1$ by how $Z_t$ is assumed to be. How do we shift this to independence of $t$?

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I dont think that that's a real problem: independence from $t-1$ is the same as independence from $t$ and you see it clearly by writing it more explicitely: for $h=1$ you simply get $Z_t+\theta Z_{t-1}\sim Z_{t+1}+\theta Z_t\quad\forall t\in\mathbb Z$ which is the same $\forall (t-1)\in\mathbb Z$. Do not get confused by dependence of the variables, stationarity is about their distribution; in fact a constant serie has dependent variables whose distribution is independent of $t$.