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
$$
\left(\frac{\bar{X}\_{t}(u)}{\sqrt{t}}\right)\_{u\in[0,1]}\xrightarrow{d} (W\_{u})\_{u\in[0,1]}\text{?}
$$



Version of Donsker-Invariance-Principle
Let $Y\_t(u):=\frac{\bar{X}\_{t}(u)}{\sqrt{t}}$ and $W(u):=W\_u$. The convergence (as $t\to\infty$) in distribution (in the Skorokhod space $D[0,1]$) of $Y\_t$ to $W$ can be proved quite similarly to the way it was done e.g. in the proof of Theorem 16.14 in 
*Foundations of Modern Probability* by Kallenberg. 


Alternatively and a bit more directly, one may use e.g. Theorem 15.6 in 
*Convergence of Probability Measures* by Billingsley (1968). Indeed, the convergence of the finite-dimensional distributions of $Y\_t$ to those of $W$ follows by the convergence of the one-dimensional distributions and the independence of the increments of the processes $Y\_t$. In view of the condition $Var\;X\_1=1$, condition (15.21) in Billingsley holds (with $Y\_t$ in place of $X\_n$) for $\gamma=2$, $\alpha=1$, and $F(u)\equiv u$ -- cf. the condition sufficient for (15.21) displayed right after Theorem 15.6 in Billingsley. 


