Let $\{X_t\}_{t\in\mathbb{N}}$ be a stationary ergodic sequence of continuous random variables with full support on the real line. Let $\lambda>1$ and $c>0$. I am interested in the probability $$ P(\forall t\in\mathbb{N},\,|X_t| \le c\lambda^t), $$ and specifically whether there are conditions to ensure that it is nonzero.
This is a generalisation of a question I asked before. There I simplified the $\{X_t\}_{t\in\mathbb{N}}$ to be iid, in which case the condition $\mathbb{E}\log|X_t|<\infty$ turns out to be sufficient for the probability to be nonzero.
Attempts:
If the probability is zero for all $c>0$, then $$ \lim_{c\rightarrow\infty}P(\forall t\in\mathbb{N},\,|X_t| \le c\lambda^t) = 0, $$ which is a contradiction, since this limit should be one. Therefore, once $c$ is sufficiently big, we know the probability is nonzero.
Next, I investigate the claim for any $c>0$. By the expectation condition one could get $$ \sum_{t=1}^{\infty} \mathbb{P}(X_t>c\lambda^t) \approx \mathbb{E}\log|X_t| < \infty, $$ so that by Borel-Cantelli $\mathbb{P}(\liminf\{\omega\in\Omega\mid |\epsilon_{k}| \le c\lambda^k\}) = 1$. Therefore there exists a random variable $N\in\mathbb{N}$ such that $|\epsilon_{k}| \le c\lambda^k$ for all $k\ge N$ and thus we would have $$ P(\forall t\in\mathbb{N},\,|X_t| \le c\lambda^t) = P(\forall t<N,\,|X_t| \le c\lambda^t). $$ I'm not sure whether I am allowed to conclude that this last probability is nonzero.