Stationarity of AR(p) process I'm looking for a proof for the stationarity of an AR(p) process, I know a stationary process $Yt$ must fulfill the following conditions:
$(1)$-$E(Y_t)=m$ for all values of $t$.
$(2)$-$Var(Y_t)=\sigma^2$ for all values of $t$.
$(3)$-$cov(Y_t,Y_s)=\gamma(|t-s|)$ for every $t$ and $s$.
First I'll define an AR(p) process $Y_t$ as follows:
$\phi(B)Y_t=\epsilon_t$ with: $BY_t=Y_{t-1}$, $\epsilon_t$ a white noise and
$\phi(B)=1-\alpha_1B-...-\alpha_pB^p$.
The condition that I read about in several posts is:
$(4)$ If the modulus of each root of $\phi$ is strictly greater than 1 then the AR(p) process as defined before is stationary.
So, what I'm looking for is a proof for: $(4)$$<=>$$(1)$&$(2)$&$(3)$
 A: We first start with AR(1) with $|\phi|<1$. We write $$x_t=\phi^{-1}x_{t-1}+w_{t}=\phi(\phi x_{t-2}+w_{t-1})+w_{t}=\dots=\phi^{k}x_{t-k}+\sum_{j=1}^{k-1}\phi^{j}w_{t-j}$$ We assume $sum_t var(x_t)< \infty$, and we can write $$x_t=\sum \phi^j w_{t-j}$$ now$$E(x_t)=\sum \phi^j E(w_{t-j})=0 $$ autocovariance function is 
$$cov(x_{t+h},x_t)=E((\sum_{j}\phi^j w_{t+h-j})(\sum_k\phi^k w_{t-k}) )=E((w_{t+h}+\dots+\phi^h w_t+\phi^{h+1}w_{t-1}+\dots))\\(w_t+\phi w_{t-1}+\dots)=\sigma^2_w\sum_j \phi^{h+j}\phi^j=\sigma^2_w\phi^h\sum_j\phi^{2j}=\frac{\sigma_w^2\phi^k}{1-\phi^2} $$
The other way:
$$x_t=\phi x_{t-1}+w_t\rightarrow(1-\phi B)x_t=w_t\rightarrow1-\phi B=0\rightarrow x=\frac{1}{\phi}\rightarrow |\frac{1}{\phi}|>1\rightarrow|\phi|<1$$
When $|\phi|>1$, the process is called explosive and one can prove that every explosive process has a stochastically equivalent causal process. 
To generalize for AR(p):
$$x_t=\phi_1x_{t-1}+\phi_2x_{t-2}+\dots+\phi_px_{t-p}+w_t$$ We rewrite this as an AR(1) model $y_t=Fy_{t-1}+w_t$
[
\begin{bmatrix}
    x_{t} \\
    x_{t-1}      \\
    \dots\\
    x_{t-h+1}      
\end{bmatrix}=\begin{bmatrix}
    \phi_{1} & \phi_{2} & \phi_{3} & \dots  & \phi_{p} \\
    1 & 0 & 0 & \dots  & 0 \\
    \vdots & \vdots & \vdots & \ddots & \vdots \\
    0 & 0 & 0 & \dots  & 1
\end{bmatrix}\begin{bmatrix}
    x_{t-1}       \\
    x_{t-2}       \\
    \dots    \\
    x_{t-p}       
\end{bmatrix}+\begin{bmatrix}
    w_{t}       \\
    0       \\
    \dots    \\
    0       
\end{bmatrix}
]
Lets call thus augmented AR(1) notation. We can write the AR model recursively $$y_t=F^ty_0+w_t+Fw_{t-1}+F^2w_{t-2}+\dots+F^{t-1}w_1+F^tw_0$$. Recall from eigenvalue decomposition: $F=Q\Lambda Q^{-1}$ and $F^j=Q\Lambda^jQ^{-1}$ where $\Lambda^j$ is
\begin{bmatrix}
    \lambda_{1}^j       & 0 & 0 & \dots & 0 \\
    0      & \lambda_{2}^j & 0 & \dots & 0 \\
    \dots \\
    0       & 0 & 0 & \dots & \lambda_p^j
\end{bmatrix}
and is stable when $|\lambda_i|<1$ for all i.
The actual eigenvalues of F satisfy $$\lambda^p-\phi_1\lambda^{p-1}-\phi_2\lambda^{p-2}-\dots-\phi_{p-1}\lambda-\phi_p=0$$ So the eigenvalues are the reciprocal of the values that solve the characteristic polynomial of the AR(p), so the roots should be outside the unit circle 
