Suppose $(X_n)_{n \in \mathbb Z}$ is an ergodic sequence (where $X_i$'s are real valued random variables). I want to answer the following question:
- Is the sequence $(\ldots,X_{-2}+X_{-1}, X_0 + X_1, X_2 + X_3, \ldots)$ also ergodic?
But I know from Samorodnitsky's "Stochastic Processes and Long Range Dependence" that the following sequence is ergodic:
$(\ldots,X_{-2}+X_{-1}, X_{-1}+X_{0}, X_0 + X_1, X_1+X_2, X_2 + X_3, \ldots)$ is ergodic. So question 1. reduces to answering the following question:
- Is $(X_{2n})_{n \in \mathbb Z}$ ergodic?
My attempts:
I tried considering some examples and in both the (simple) examples I considered, $(X_{2n})_{n \in \mathbb Z}$ also turned out to be ergodic:
Ex1. $X_i$ are iid random variables, then clearly true.
Ex2. $(X_n)_{n \in \mathbb Z}$ is a Markov chain which is aperiodic, irreducible and positive recurrent then $(X_{2n})_{n \in \mathbb Z}$ is also a Markov chain having all the above mentioned properties and hence is ergodic.