I have a $\mathbb{Z}_+$-valued stochastic process $X$ in discrete time, which has unit jumps up or down. I know the following statement: there exists a random variable $\tau$, almost surely finite, s.t. the time between successive returns of $X$ to 0 is stochastically bounded by $\tau$. That is, if $T_i$ is the time between $i$th and $(i+1)$th visits to 0, then $P(T_i>n) \leq P(\tau > n)$ for all $i$. (The process isn't Markov, so there isn't any reason to think that $T_i$ are iid.)

I would like to derive some statement of the form: for any $\epsilon > 0$ there exists $K < \infty$ such that $\lim\inf_{T \to \infty} \frac1T \sum_{n=0}^T \delta_{X_n < K} \geq 1-\epsilon$. This seems plausible, because morally $X$ should be upper bounded by $\tau/2$ (it has unit jumps, and in time $\tau$ needs to return to 0), but I'm having trouble formalizing this intuition.