Assume we have a Levy process $(X_t)_{t\geq 0}$ with a finite second moment for all $t>0$. For simplicity, say $\operatorname{Var}\left[X_1\right]=1$. Let $\tilde{X}_t:=X_t-t\cdot E\left[X_1\right]$. Define $\bar{X}_t(u):=\tilde{X}_{tu}$. Can we say, that for fixed $u\in[0,1]$

$$ \frac{\bar{X}_{t}(u)}{\sqrt{t}}\xrightarrow{d} W_{u} $$ as $t \rightarrow \infty$, where $W$ is a standard Wiener process?

More generally, do we have the convergence $$ \left(\frac{\bar{X}_{t}(u)}{\sqrt{t}}\right)_{u\in[0,1]}\xrightarrow{d} (W_{u})_{u\in[0,1]}\text{?} $$