I have difficulties to understand this in ergodic theory (approach form proabrbility theory).
Usually, an ergodic process is defined by:
$$
X_n(\omega) := X(\phi^n(\omega)),
$$
where $\phi$ is a measure preserving transformation. (And the invariant $\sigma$-field is trivial.)
I think I do have a pretty good understanding what a measure preserving function is and what it does. However, I am absolutely clueless, how I should understand the definition of $X_n$. Is it not the case, that there is a full dependency between the states?
I tried to model an example:
Consider the Markov Process $X_i$ on $\Omega:= \lbrace 0,1 \rbrace$ with values in $\lbrace 0,1 \rbrace$ and $P(X_{i+1} = x \vert X_i = y) := p(y,x)$, where $p(x,y) = \frac{3}{4} $ for $x \neq y $ and $p(x,x) = \frac{1}{4}$. (This chain is ergodic since it is aperiodic and irreducible.)
To my understanding, the only measure preserving transformation is the identity, since 0 and 1 have different wights. But that would imply that $X_2$ is equal to $X_1$?
Thanks for taking your time!
Best Answer
The issue is that you have the incorrect probability space. Remember, your space $\Omega$ needs to be rich enough to deal with sample paths, not just sample points. For example, $(X_1,X_2,X_3) = (0,0,0)$, $(0,1,0)$, and $(0,1,1)$ should all have positive probability for the Markov chain you have defined, and yet your probability space only has two points! The correct probability space to consider is
$$\Omega :=\{0,1\}^{\mathbb Z}$$
with the product $\sigma$-field, and the unique probability measure $\mathbb P$ satisfying
$$\mathbb P\Big(\big\{\omega=\{\omega_i\}_{i\in\mathbb Z}\in\Omega:\omega_i = x_i\text{ for }i\in\{n,n+1,\ldots,n+m\}\big\}\Big) = \mu(x_n)\prod_{i=1}^mp(x_{i-1},x_i)$$
for each $n\in\mathbb Z$, $m\in\mathbb N$, and $(x_n,\ldots,x_{n+m})\in\{0,1\}^{m+1}$, where $\mu = \frac12\delta_{0} + \frac12\delta_{1}$ is the invariant distribution of your Markov chain. Your measure preserving transformation $\phi$ is just the shift operator, i.e.
$$\phi\Big(\{x_i\}_{i\in\mathbb Z}\Big) = \{y_i\}_{i\in\mathbb Z} \quad \text{where }y_i=x_{i+1}$$
and your observable $X$ is the just the value of the $0$ component of your sample point $\omega$, i.e.
$$X\Big(\{\omega_i\}_{i\in\mathbb Z}\Big) := \omega_0.$$
Under this notation, $X_n(\omega)=\omega_n$, so $\{X_n\}$ is a Markov chain with initial distribution $\mu$ and transition probabilities given by $p$. By construction, this is an ergodic Markov chain, and $\phi$ is an ergodic transformation.
More generally, if you replace $\{0,1\}$ with a countable state space $E$ and let $p$ be a transition kernel on $E$ with invariant distribution $\mu$, then the measure preserving transformation $\phi$ will be ergodic if and only if the induced Markov chain is ergodic, so the (seemingly different) definitions are consistent.