Stationary ergodic process with $X_n/f(n) \overset{a.s.}{\nrightarrow} 0$?

Question

I am searching for a stationary ergodic process $(X_n)$ of nonnnegative random variables with finite expectation, as well as a sequence $f(n)\uparrow \infty$ such that $$ X_n/f(n) \overset{a.s.}{\nrightarrow} 0, $$ i.e. almost sure convergence does not hold.

Answer 1

Let $X_n$ be independent exponential(1) random variables, i.e., $P(X_n>x)=e^{-x}$ for all $x \ge 0$. Let $f(n)=\ln(n)$. Then $P(X_n>f(n))=1/n$ forms a divergent series, so by the second Borel Cantelli Lemma, with probability 1 there are infinitely many $n$ such that $X_n>f(n)$.

https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma#Converse_result