Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Question

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\mbox{ a.s.} $$ Is there any similar result for the $\limsup_n$ or $\liminf_n$ of the sequence $\{f(X_k)\}$, i.e. results which involve expectation? More specifically is there a way to find $$\limsup_n f(X_n),\ \liminf_n f(X_n)$$ almost surely? Though I have asked the question for the general ergodic processes, even results for stationary processes like Gaussian processes will be helpful to me. Also, if they ar enot available, it will be very kind if someone can give me some references that I can use to find methods to find these results. Thanks in advance.

Answer 1

The question was asked in this Math Overflow thread, which received an answer. An explicit form for $\limsup_n f(X_n)$ in terms of the distribution of $f\left(X_1\right)$ was given.