Reference that AR(p) are strictly stationary if and only if it is causal I am interested in conditions that an autoregressive model AR(p) is strictly stationary. It turns out that these lecture notes https://www.bauer.uh.edu/rsusmel/phd/ec2-3.pdf (see bottom of page 13) claim that an AR(p) process is strictly stationary if and only if it is causal. Seems reasonable I think. I need a reference or proof. Can you provide either?
 A: The following isn’t exactly an answer. It’s just a few potentially helpful definitions, and an explanation of why I find your question confusing. 
If the AR polynomial $\phi(z)$ doesn't have roots on the unit circle, or in other words is stationary, then we can write the inverse of this polynomial, and multiply both sides of the model equation by $\phi^{-1}(z)$. Specifically, if this condition is met, then there exists a $\delta > 0$ such that
$$
\frac{1}{\phi(z)} = \sum_{j=-\infty}^{\infty} \chi_j z^j
$$
for $1 - \delta < |z| < 1 + \delta$. 
Causality is usually only defined for one-sided processes. So further assume that $\phi(z) = \sum_{j=0}^{\infty} \phi_jz^j$. Then the process is causal, on top of stationary, if we can further restrict the roots of $\phi(z)$ to be outside of the unit circle. So you take a process that's a function of its past values, and you can also write it in terms of past noise terms. If it's an infinite sum, the sum converges in an appropriate sense. 
Strict stationarity, on the other hand, is a statement about specific distributions of the time series. I don't see how you can prove this without further assumptions.
A sufficient condition for ergodicity was given here: Consistency of Sample Mean in Time Series Data
