# variance of autoregressive process

Let $$X_t=\alpha X_{t-1}+\varepsilon_t, t\in\Bbb N$$ be a process where $$\epsilon_t$$ are iid and follow some distribution $$F$$ with mean $$0$$ and variance $$\sigma^2$$, and $$X_0\sim G(0,\sigma^2)$$ follows some distribution $$G$$ with mean $$0$$ and variance $$\sigma^2$$, and $$|\alpha|<1$$.

I don't know how to get Var$$(X_t)=\alpha^2\sum\limits_{h=1}^t\alpha^{2h}$$

You can do it with recurrence, by putting $$X_{t-1}=\alpha X_{t-2}+\epsilon_{t-1}$$ into your equation and so on with $$X_{t-2}$$,... you obtain: $$X_t=\epsilon_t+\alpha\epsilon_{t-1}+\alpha^2\epsilon_{t-2}+\dots$$ $$Var(X_t)=\sigma^2\sum_{i=0}^{\infty} \alpha^{2i}=\frac{\sigma^2}{1-\alpha^2}$$ The last equality holds because $$|\alpha|<1$$