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In Section 3 of Page 3, the notes that one of the conditions for weak stationarity is that $\gamma_{X}(h)=Cov(X_t, X_{t+h})$, essentially that the covariance of $X_s$ and $X_t$ depend only on $t-s$ and not $t$ or $s$. If I understood this correctly, this implies that it's possible to have a sequence of random variables whose the variance increases with time and yet the sequence can still be weakly stationary? So visually the data can look more and more "spread out" with time but still be stationary?

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No. Use $h=0$ to see that the condition you quote implies constant variance over time: $$ \gamma_X(0)=\text{Cov}(X_t,X_{t+0})=\text{Var}(X_t). $$

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