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Weak stationarity states that for any two observations in time, $z_t$ and $z_{t+h}$, the correlation of the two:

depends on when the time series sample begins, $t$.

depends on the difference in time between the observations, $h$.

is non-zero.

is zero.

depends on both when the time series sample begins, $t$, and the difference in time between the observations, $h$.