My question is are all ARIMA processes also unit root processes? My guess is yes because $\{X_t\}$ is ARIMA(p, d, q) if
$(1-B)^dX_t = a(B)\epsilon_t$
is stationary ARMA(p, q).
The characteristic function for $(1-B)^dX_t = a(B)\epsilon_t$ is $(1-B)^d$, which clearly is unit root.
Also, I suppose not all unit root processes are ARIMA? Is there a good counterexample?
My third and final question is are all FARIMA processes stationary? My guess is not because it really depends on the roots of the characteristic equation. I am asking this because I see FARIMA often used as an example for a "stationary long memory" process.