For simplicity, define the AR(1) model without an intercept term, that is $$ X_t := \phi X_{t-1} + w_t $$ where $w_t\sim N(0,\sigma_w^2)$ and $w_t$ is independent of $X_{t-1}$. Also assume the time series starts from time $t=-\infty$ and evolves until time $t=+\infty$ so this rules out processes that are not really this AR(1), such as a random walk starting at 0.
Now when $|\phi|>1$:
Both lecture notes https://www.stat.berkeley.edu/~gido/ARMA-models.pdf (bottom of its page 5) and http://www.maths.qmul.ac.uk/~bb/TimeSeries/TS_Chapter4_5.pdf (its last page) say that there exists the following representation: $$ X_t = -\sum_{j=1}^\infty \phi^{-j}w_{t+j} $$ It first seems agreeable to me, because regardless of any derivation of this representation from the definition of the AR(1) model, but just suppose we start with this representation of $X_t$ in terms of $w_{t+1},w_{t+2},\ldots$ weighted by $\phi^{-1},\phi^{-2},\ldots$, the RHS does seem to converge, and such $X_t$ does satisfy the AR(1) equation $X_t=\phi X_{t-1}+w_t$.
However, this representation also seems to imply that for any index set $t_1<\ldots<t_n$ and any $h>0$, we have $$ (w_{t_1+1},w_{t_1+2},\ldots) \stackrel{d}{=} (w_{t_1+h+1},w_{t_1+h+2},\ldots) $$ thus $$ (X_{t_1},\ldots,X_{t_n}) \stackrel{d}= (X_{t_1+h},\ldots,X_{t_n+h}) $$ in other words, the explosive AR(1) time series $\{X_t\}$ is strictly stationary. Then it should be at least weakly stationary, which means the variance of each observation is constant: $$ {\rm Var}(X_t)\equiv \sigma^2 $$ for a global constant $\sigma$. However, \begin{align} \sigma^2 &= {\rm Var}(X_t) = {\rm Var}(\phi X_{t-1}+w_t) = \phi^2{\rm Var}(X_{t-1}) + {\rm Var}(w_t)\\ &= \phi^2\sigma^2 + \sigma_w^2 > \sigma^2 \end{align} Then I found some other lecture note https://www2.stat.duke.edu/courses/Fall99/sta290/Notes/AR/ar.pdf (first paragraph) which says explosive AR(1) is non-stationary.
I'm deeply confused, because both "stationary" and "non-stationary" seemed to be supported by (at least seemingly) solid evidences... Where is the error? Can someone help with some clarifications? Many thanks!
I perhaps figured it out myself... Please correct me if I'm wrong.
The real problem is who is independent of whom. The representation of $X_t$ in terms of $w_{t+1},w_{t+2},\ldots$ is wrong because it does not satisfy the AR(1) equation's required independence condition: AR(1) equation requires that $X_{t-1}$ to be independent of $w_t$, however, in that representation, $X_{t-1}$'s expansion includes $w_t$ as the first term and thus ${\rm Cov}(X_{t-1},w_t) = {\rm Cov}(-\phi^{-1}w_t,w_t) \neq 0$ -- the independence requirement is violated.