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In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting:

  1. $(\Omega,\mathcal F,\mathsf P)$ is a probability space,

  2. $X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random element and

  3. $\varphi:(\Omega,\mathcal F)\to (\Omega,\mathcal F)$ is a measure-preserving map, i.e. for any $A\in \mathcal F$ it holds that $$\mathsf P(\varphi^{-1} A) = \mathsf P(A).$$

The event is called invariant if $\mathsf P((\varphi^{-1}A)\Delta A) = 0$. I am interested in some examples for such events since books I'm reading have a very few of them.

If we put $X_n(\omega) = X(\varphi^n \omega)$ then we obtain a stochastic process on $S$. By the way, is it true that $X$ is always a stationary Markov process?

An event
$$ A = \{X_n\in B\text{ infinitely often}\} $$ is invariant as well as its complement $$ A^c = \{\text{ there exists }N \text{ such that } X_n \in B^c \text{ for all }n\geq N\}. $$

Moreover, all invariant events form a $\sigma$-algebra and if $X$ is Markov then its invariant event are characterized by its harmonic functions. However, these characterization are quite elusive, so for me it is difficult to imagine other invariant events based on such characterization.

I would be also grateful if you can refer me to a book/lecture notes where such examples are provided.

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up vote 2 down vote accepted

probably you already know this example, but in any case...

let $\Omega=\{0,1\}^{\mathbb{N}}$, $\mathcal{F}$ the $\sigma$-algebra generated by the cylinder sets of $\{0,1\}^{\mathbb{N}}$ and $ \mathbb{P}=\prod_{i\in\mathbb{N}} \mu_i $ is product measure with $\mu_i:\mathcal{P}(\{0,1\})\to [0,1]$ given by $\mu_i(\{0\})=\frac{1}{2}=\mu_i(\{1\})$ for all $i\in\mathbb{N}$.

We define $\varphi:\Omega\to\Omega$ in the following way:
We think about an element $\omega\in\Omega$ as an infinite sequence of zeros and one, that is, $\omega=(\omega_1,\omega_2,\ldots)$ and then put $$ \varphi(\omega_1,\omega_2,\ldots)=(\omega_2,\omega_3,\ldots) $$ this function is known as the left-shift.

You can prove that $\varphi$ is a measure-preserving map. The trick is prove that $\mathbb{P}(\varphi^{-1}\mathscr{C})=\mathbb{P}(\mathscr{C})$ for any cylinder set and then extend this to the whole $\sigma$-algebra using for example the extension measure theorems.

Aside comment: We can prove a more strong fact, this product measure is in fact ergodic for the shift, which implies that the invariant sets has measure zero or one.

To finish the example take $A=\Omega\setminus\{1,1,1,\ldots\}$ which is a measure one set. Note that $\varphi^{-1}A=\Omega$, once for any $\omega\in\Omega$ we have $\mathbb{P}(\{\omega\})=0$ (the proof of this statement is a consequence of the continuity of $\mathbb{P}$), follows that $\mathbb{P}(\varphi^{-1}A\Delta A)=0$.

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First of all, note that any stationary process ${(X_n)}_{n \in \mathbb{Z}}$ can be represented as $X_n(\omega)=X(\phi^n\omega)$. It suffices to take the shift $\phi$ on the canonical space of the process (see here) and $X$ which maps a sequence to its central coordinate.

The invariant $\sigma$-field is then the tail $\sigma$-field $\cap_{n \leq 0}\sigma(X_m, m\leq n)$.

If $A$ is a recurrence class of a Markov $(X_n)$ then it is easy to see that the event $\{X_n\in A\}$ belongs to the invariant $\sigma$-field.

But the irreducibility does not guarantee that the invariant $\sigma$-field is degenerate. Consider for instance a stationary random walk on a square $ABCD$. At time $n$ the random variable $X_n$ is uniform on the vertices of the square and conditionally to $X_n$ the random variable $X_{n+1}$ is one of the two vertices connected to $X_n$. This Markov chain is irreducible but in the tail $\sigma$-field we know something about the process: the event $\big\{X_{2n} \in \{A,C\} \text{ for all $n$}\big\}$ belongs to the tail $\sigma$-field. Here the non-degeneracy of the tail $\sigma$-field is related to the periodicity of the Markov chain.

Ergodicity of $\phi$ is equivalent to irreducibility of the Markov chain, whereas the degeneracy of the tail $\sigma$-field is equivalent to the strongest property called $K$-property.

See also this topic for your interest.

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