[Math] $\sigma$-algebra of $\theta$-invariant sets in ergodicity theorem for stationary processes

Question

Applying Birkhoff's ergodic theorem to a stationary process (a stochastic process with invariant transformation $\theta$, the shift-operator – $\theta(x_1, x_2, x_3,\dotsc) = (x_2, x_3, \dotsc)$) one has a result of the form

$ \frac{X_0 + \dotsb + X_{n-1}}{n} \to \mathbb{E}[ X_0 \mid J_{\theta}]$ a.s.

where the right hand side is the conditional expectation of $X_0$ concerning the sub-$\sigma$-algebra of $\theta$-invariant sets… How do these sets in $J_{\theta}$ look like? (I knew that $\mathbb{P}(A) \in \{0,1\}$ in the ergodic case, but I don't want to demand ergodicity for now).

Answer 1

One asks that $A$ is such that $(x_n)_{n\geqslant1}\in A$ if and only if $(x_{n+1})_{n\geqslant1}\in A$. The surprising fact is that such events $A$ do exist, whose definition is not trivial, and in fact a lot of them. For example, $A$ is invariant as soon as the fact that $(x_n)_{n\geqslant1}\in A$ depends only on:

• the liminf and/or the limsup of $\frac1{b_n}\sum\limits_{k=1}^na_ku(x_k)$ when $n\to\infty$, for some function $u$ and some sequences $(a_n)_{n\geqslant1}$ and $(b_n)_{n\geqslant1}$ such that $b_n\to\infty$.
• the liminf and/or the limsup of $\frac1{b_n}\sum\limits_{k=1}^na_ku(x_k,\ldots,x_{k+N})$ when $n\to\infty$, for some fixed $N$, some function $u$ and some sequences $(a_n)_{n\geqslant1}$ and $(b_n)_{n\geqslant1}$ such that $b_n\to\infty$.
• the liminf and/or the limsup of $\frac1{b_n}\sum\limits_{k=1}^na_ku((x_{k+i})_{i\geqslant0}))$ when $n\to\infty$, for some function $u$ and some sequences $(a_n)_{n\geqslant1}$ and $(b_n)_{n\geqslant1}$ such that $b_n\to\infty$.
• the fact that $\{n\geqslant1\,;\,x_n\in B\}$ is finite or infinite, for some $B$.
• the fact that $\{n\geqslant1\,;\,(x_n,\ldots,x_{n+N})\in B\}$ is finite or infinite, for some fixed $N$ and some $B$.
• the fact that $\{n\geqslant1\,;\,(x_{n+i})_{i\geqslant0}\in B\}$ is finite or infinite, for some $B$.