Consider an infinite sequence $X_1,\cdots,X_t$ of i.i.d. Bernoulli random variables with parameter $p \in (0,1)$. Consider also an infinite sequence $\alpha(t)$ of integers such that $\alpha(t) \leq t$ for all $t$ and \begin{equation*} \lim_{t \rightarrow +\infty}\dfrac{\alpha(t)}{t} = p \end{equation*} I am trying to show that, with strictly positive probability \begin{equation*} \sum_{i=1}^{t}\dfrac{X_i}{t} \geq \dfrac{\alpha(t)}{t} \end{equation*} for all $t \in \mathbb{N}$.
The interpretation is as follows. The process stops whenever the number of successes up to date $t$ is lower than $\alpha(t)$. The frequency of successes required to continue converges to the true frequency of the data-generating-process when $t$ becomes large. I want to show that with strictly positive probability this process never stops.
There is a related well-known result in the literature on martingales: the process that stops whenever the frequency of successes falls below $p$ survives forever with positive probability. I am trying to generalize it with a non-constant threshold. Any hint ot reference? Thank you!