Is every set of i.i.d random variables ergodic?

Question

sorry if this is a simple question, but I'm not sure: Is every set of i.i.d random variables automatically ergodic or do we need some extra conditions for that? Thank you

Answer 1

The statement is true.

W.l.o.g. let your i.i.d. r.v. $(X_n)_{n \in \mathbb{N}}$ be the canonical process on $\Omega$. Your dynamical system is then given by $(\Omega,\mathcal{F},P, \tau)$, where $\tau$ is the shift-operator and $\mathcal{F}$ is the $\sigma$-field generated by your process. We need to check that an invariant set $A \in \mathcal{F}$ is trivial. Notice that $A = \tau^{-k} (A) \in \sigma(X_k,X_{k+1},...)$ for arbitrary $k \in \mathbb{N}$. This implies that $A$ is in the terminal $\sigma$-field. As your $(X_n)_{n \in \mathbb{N}}$ are independent we get that $A$ is trivial.