I am reading a book, which uses the Central Limit Theorem of Levy Processes $X_{t}$ without mentioning the exact theorem. Due to the infinite divisible property i can write $X_{t}$ as a sum of iid random variables $X^{i}$ \begin{align} X_{t}=\sum_{i=1}^{N}X^{i}_{t/N} \end{align} The Problem is, that i want $t\rightarrow \infty$ but for the CLT i have to keep my sequence of $N$ iid random variables fixed. But they do change. The book now just says, that with the central limit theorem for Levy Processes it holds for $t\rightarrow \infty$ \begin{align} \frac{X_{t}-\overbrace{tE[X_{1}]}^{=E[X_{t}]}}{\sqrt{t}}\rightarrow \mathcal{N}(0,Var[X_{1}])\\ \sqrt{t}(\frac{X_{t}}{t}-E[X_{1}])\rightarrow \mathcal{N}(0,Var[X_{1}]) \end{align} I cant find any proofs, lectures or literature about it. Can you help me out?