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I'm a bit confused about the definition of weakly stationarity

from looking at the definition of weakly stationarity, it requires: E(xt) = E(xt−j ) = µ ∀ j var(xt) = var(xt−j ) = σ^2 ∀ j cov(xt, xt−j ) = γj ∀ t

But from looking at the graph of causal AR(1) + drift + trend, I don't think the E(xt) is constant (mean is increasing as t increases), also the variance is not constant from the graph, so why is causal AR(1) + drift + trend weakly stationary? enter image description here

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  • $\begingroup$ Hi: if you add a trend to a weakly stationary series ( in other words, a $b \times t $ term ), then the process could be termed trend stationary which means that, if you de-trended, you'd get a weakly stationary process. But, I don't see how you can tell that the variance is not constant ? Also, it would be better if you wrote out the whole equation so that it was clear what you meant by drift ? Are you just adding a constant to the process ? $\endgroup$
    – mlofton
    Commented Jul 5, 2021 at 5:49

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