Limit of stochastic subsequence of stationary ergodic sequence

Question

Let $\{X_k\}_{k\in\mathbb{N}}$ be a stationary ergodic sequence on a probability space $(\Omega,\mathcal{F},P)$ with shift $T$. Also, let $\{v_k\}_{k\in\mathbb{N}}$ be a sequence of random variables in $\mathbb{N}$ such that $v_k$ is a function of $X_1,\ldots,X_{v_k-1}$ and $v_k>v_{k-1}$. Suppose $A\in\mathcal{F}$ is an event with $P(A)>0$.

I am trying to investigate the quantity $$ \limsup_{k\rightarrow\infty}P(T^{v_k}A), $$ and specifically whether extra assumptions are needed to ensure this quantity is nonzero.

In the case that $\{X_k\}_{k\in\mathbb{N}}$ is iid one can use \begin{align} P(T^{v_k}A) = \sum_{\ell=1}^{k}P(T^{v_k}A\mid v_k = \ell)P(v_k=\ell) \stackrel{!}{=} \sum_{\ell=1}^{k}P(T^{\ell}A)P(v_k=\ell) = P(A) > 0, \end{align} however with only the SE assumption I am stuck, since the conditioning on $v_k=\ell$ cannot be ignored at !. An alternative approach I am thinking of is using some form of LLN to get $$ \lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n} 1_{\{T^{v_k}A\}} = P(A) > 0, $$ which would imply that the limsup is nonzero. However, this does not necessarily seem to hold. I am quite unexperienced at working with stationary ergodic sequences, any help is highly apreciated!

Answer 1

Mixing may be sufficient (not sure).

Recall that a transformation is mixing if $\mu(T^{-k}A \cap B) \rightarrow \mu(A)\mu(B)$ as $k \rightarrow \infty $ for all $A$, $B$.

If $T$ is just weak-mixing, we still have $\mu(T^{-k}A \cap B) \rightarrow \mu(A)\mu(B)$ but only along a subsequence of density one, so I guess it does not hold.

In the ergodic case, I would try to build a counterexample by using an irrational rotation $R_\theta$ of the circle (identify the circle with $[0,1[$ and take $X_k(x) = R^k_\theta(x)$. Note that $X_1$ is just the identity). Take for $A$ some interval containing $0$ and cut it into subintervals of size $1/k$. Define $v_k$ on each of these subintervals so that under $T^{v_k}$, they end close to $[0,1/k[$. So they stack at time $v_k$, $A$ is sent close to $[0,1/k[$ by $T^{v_k}$ and the limsup should be 0. Note that $v_k$ depends only on $x=X_1(x)$.