I want to discuss the Central Limit Theorem (CLT) of a Lévy process under some assumptions. The answer of the previous post Central Limit Theorem for Lévy Process required a wide base of knowledge, without any assumptions on the Lévy process. So I'm considering now a more specific setup:
Setup: Given a Lévy process on $\mathbb{R}^d$ with $E|X_1|<\infty$ (thus $E|X_t|<\infty$ for $t>0$) and finite second moment for all $t\geq 0$. Let $\Sigma$ be the positive definite covariance Matrix of $X_1$. Then for $n:=\lceil t \rceil$ or $n:=\lfloor t \rfloor$ then $$ X_n=\sum_{i=1}^n X_1 $$ is considered (due to the Lévy property) as a partial sum of $n$ i.i.d. variables. Then by CLT we have $$ \sqrt{n}\left(\frac{X_n}{n}-E[X_1]\right) \rightarrow \mathcal{N}\left(0,\Sigma\right) \text{ as } n\rightarrow \infty $$ with covariance Matrix $\Sigma$.
From this I want to conclude, that the CLT holds for real values $t$ also.(so I may avoid the proof a new CLT in continuous time for this given setup i.e. by characteristic functions). So \begin{align} \sqrt{t}\left(\frac{X_t}{t}-E[X_1]\right) \rightarrow \mathcal{N}\left(0,\Sigma\right) \text{ as } t \rightarrow \infty \end{align}
My idea Reduce the setting to the classical CLT by splitting up. Choose $n:=\lfloor t \rfloor$ \begin{align} \frac{X_t}{t}=\frac{n}{t}\left(\frac{X_n}{n}+\frac{X_t-X_n}{n}\right) \tag 1 \end{align}
\begin{align} \sqrt{t} \left(\frac{n}{t}\left(\frac{X_n}{n}+\frac{X_t-X_n}{n}\right)-E[X_1]\right) \end{align} Define $U_n:=\frac{1}{n} \sup_{t\in [n,n+1]}|X_t-X_n|$. $U_n$ is i.i.d. Since $E|X_{1}|<\infty$ it follows that $U_1$ has finite expectation, see Finite supremum. By classical Law of Large Number we have \begin{align} \lim_{n\rightarrow \infty} n^{-1}\sum_{i=1}^n U_{n}\rightarrow E[U_1]\quad P\text{-a.s.} \end{align} such that $n^{-1}U_{n}\rightarrow 0$ $P\text{-a.s.}$ (as $n\rightarrow \infty$ by $t\rightarrow \infty$) which implies convergence in distribution. Note that $U_n$ dominates the second term in the brackets in $(1)$.
Then we can conclude from that, that (remark that $n:=\lfloor t \rfloor$) $$ \lim_{t\rightarrow \infty}\frac{X_t}{t}=\lim_{t\rightarrow \infty}\frac{n}{t} \frac{X_{n}}{n}=\lim_{n\rightarrow\infty}\frac{X_{n}}{n}\quad P\text{-a.s.} $$ Now sandwich for $n:=\lfloor t \rfloor$ $$ \sqrt{n}\left(\frac{n}{t}\left(\frac{X_n}{n}+\frac{X_t-X_n}{n}\right)-E[X_1]\right)\leq \sqrt{t}\left(\frac{n}{t}\left(\frac{X_n}{n}+\frac{X_t-X_n}{n}\right)-E[X_1]\right) \leq \sqrt{n+1}\left(\frac{n}{t}\left(\frac{X_n}{n}+\frac{X_t-X_n}{n}\right)-E[X_1]\right) $$ Where the upper and lower bound converges to $\mathcal{N}(0,\Sigma)$ as $t\rightarrow \infty$.
For three random-variables $X_n,X_n',Y$ with $X_n$ and $X_n'$ converge in distribution according to a random variable Z. If $X_{n}\leq Y\leq X_n'$, then $Y\rightarrow Z$ in distribution.
