Let $(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t \ge 0})$ be a filtered probability space. On it I have defined an $\mathcal{F}_{t}$-adapted process $(Y_{t})_{t \ge 0}$. It is a a continuous parameter, pure-jump process, uniformly bounded in $[0,1]$. Let $(T_{k})_{k \ge 0}$ be a sequence of $\mathcal{F}_{t}$-stopping times satisfying the properties:
1-. $\lim_{k \to \infty} T_{k} = +\infty$ a.s.
2-. For each $k \ge 0$, $T_{k}$ is finite a.s.
3-. On the set $\{T_{k} \le t < T_{k+1}\}$, we have that $Y_{t} \ge Y_{T_{k}}$
(in the context I am dealing with, the $T_{k}$'s are the renewal times of a Poisson process, an so the first two conditions hold; as for the third condition, it is very problem-specific).
I know that:
\begin{align} \lim_{k \to \infty}\mathbb{E}(Y_{T_{k}})=1 \end{align}
By looking at the third condition above, and considering the fact that $(Y_{t})_{t \ge 0}$ is uniformly bounded above by $1$, it seems rather natural to guess that: \begin{align} \lim_{t \to \infty}\mathbb{E}(Y_{t})=1 \end{align}
but I've been unable to prove it.
I'd be very grateful if anyone could help me to prove or disprove this claim.
EDIT: It is possible to consider a fourth condition on the stopping times.
4-. For each $k\ge 0$, $T_{k+1}-T_{k}$ is independent of the stopped $\sigma$-field $\mathcal{F}_{T_{k}}$.
With this, I can proceed as follows. Let $\delta > 0$. Fix $k_{0}>0$ s.t. $\mathbb{E}(Y_{T_{k}})>1-\delta$ for every $k \ge k_{0}$. Then:
$$\mathbb{E}(Y_{t}) \ge \sum_{j \ge k_{0}}\mathbb{E}(Y_{t} \mathbf{1}_{\{t \in [T_{j}, T_{j+1})\}}) \ge \sum_{j \ge k_{0}}\mathbb{E}(Y_{T_{j}} \mathbf{1}_{\{t \in [T_{j}, T_{j+1})\}})= \sum_{j \ge k_{0}}\mathbb{E}(Y_{T_{j}} \mathbf{1}_{\{ T_{j} \le t\}})\mathbb{P}(t \in [T_{j}, T_{j+1})) \ge (1-\delta)\mathbb{P}(t \ge T_{k_{0}})- \sum_{j \ge k_{0}} \mathbb{P}(T_{j} \ge t) \mathbb{P}(t \in [T_{j}, T_{j+1}))$$
but I don't manage to go further this point.