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During my studies I came across the $I(g)$-symbol, in the sense that $Z_t \sim I(g)$ for an integer $g$ and $Z_t$ is a times series. What exactly does this mean, and does it have to do with stationarity?

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It's hard to tell without more context. But mostly sure it means "$Z_t$ is integrated of order $g$", i.e. $\Delta ^g Z_t=(1-d)^gZ_t$ (where $d$ is the lag operator) is stationary, in other words you have to difference $Z_t$ $g$ times to get an stationary time series. See this.

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