I have a continuous-time stochastic process $X=\left(X_{t}\right)_{t \geq 0}$ for which I showed that there is $v>0$ such that, for integer times,
$$\frac{X_n}{n} \to v, \,\, \text{ $\mathbb{P}-a.s.$}$$
I can also show that \begin{eqnarray}\label{1} \mathbb{P}\left(|X_{t}-X_{\lfloor t \rfloor}|\geq\epsilon t\right) \leq \alpha e^{-\beta t}, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\ (1) \end{eqnarray} for every $\epsilon > 0$, where $\alpha$ and $\beta$ are constants that depend on $\epsilon$.
I want to show that
$$\frac{X_t}{t} \to v, \,\, \text{ $\mathbb{P}-a.s.$}$$
For this, I thought of writing the following:
$$\frac{X_{t}}{t}=\frac{X_{t}-X_{\lfloor t \rfloor}+X_{\lfloor t \rfloor}}{t}=\frac{X_{t}-X_{\lfloor t \rfloor}}{t}+ \frac{X_{\lfloor t \rfloor}}{t}.$$
But if I use Inequality $(1)$ I am going to get convergence in probability to $0$ for the first term on the right side, which does not imply almost surely convergence. Also, the second term should be divided by $\lfloor t \rfloor $ and not by $t$. Does anyone know a standard way of procedure in this case?