The characteristic function of $X_t$ in a AR(1) process

Question

Let $(X_t)_{t\in \mathbb{Z}}$ an ergodic and stationary $AR(1)$ process. That is, let $$X_t = \phi X_{t-1} + \epsilon_t$$ where $(\epsilon_t)$ is an iid noise sequence. Can I say that the characteristic function of $X_t$ is the same for every $t$? That is: $$\varphi_{X_{t_1}}= \varphi_{X_{t_2}}, \quad \forall t_1 \neq t_2 $$ Some help?

Answer 1

Yes, the characteristic functions of $X_{t_1}$ and $X_{t_2}$ are the same; this follows because $X_t$ is in fact a strongly stationary process, so that all the marginal distributions are the same. This is a special case of the fact that linear processes driven by strong white noise are strongly stationary.

Provided that $|\phi|<1$, an $AR(1)$ process has the linear representation $$X_t = \sum_{k=0}^\infty \phi^k \epsilon_{t-k}$$ with respect to the iid noise $(\epsilon_t)$. It follows from the theorem cited above that $X_t$ is strongly stationary and thus all the margins have the same characteristic function.