Why is this process a Levy process
Hello. I am currently studying Levy processes. In particular, I am trying to understand the proof of the following Theorem:
" Let $b \in \mathbb{R}^d$, $M \in \mathbb{R}^{d \times d}$ non-negative definite, let $\lambda$ be a measure on $\mathbb{R}^d \setminus \{0\}$ such that $$\int \text{min}(1,|x|^2) \lambda(\text{dx}) < \infty$$ For $\theta \in \mathbb{R}^d$ set $$\Psi (\theta) = i \langle b, \theta \rangle - \frac{1}{2} \langle \theta , M \theta \rangle + \int_{\mathbb{R}^d} \left( e^{i \langle \theta, x \rangle} - 1 - i \langle \theta, x \rangle 1_{\{|x| \leq 1\}} \right) \lambda(\text{dx})$$ Then there exists a Levy process $(X_t)_{t \geq 0}$ such that its characteristic exponent is given by $\Psi$. "
Unfortunately, there is only a sketch of the construction given, the idea is somewhat like this: We construct such a process by considering processes $X^{(1)}, X^{(2)}, X^{(3)}$. These processes are defined as:
\begin{align} X_t^{(1)} &= \sqrt{M} B_t + bt \\ X_t^{(2)} &= \int_{0}^t \int_{\mathbb{R}^d} x \mathfrak{P}^{(2)}(\text{ds}, \text{dx}) \\ X_t^{(3)} &= \lim_{\varepsilon \rightarrow 0} \int_{0}^t \int_{\mathbb{R}^d} x1_{\{\varepsilon < |x| \leq 1\}} \mathfrak{P}^{(3)}(\text{ds}, \text{dx}) - t \int_{\mathbb{R}^d} x1_{\{\varepsilon < |x| \leq 1\}}\lambda(\text{dx}) \end{align}
where $\mathfrak{P}^{(2)}$ and $\mathfrak{P}^{(3)}$ are the Poisson Point processes for the intensity measures $\text{dt} \times \lambda^{(2)}$ and $\text{dt} \times \lambda^{(3)}$, respectively, where $\lambda^{(2)} (\text{dx}) = \lambda(\text{dx}) \cdot 1_{\{|x| > 1\}}$ and $\lambda^{(3)} (\text{dx}) = \lambda(\text{dx}) \cdot 1_{\{|x| \leq 1\}}$. It is claimed that these processes are Levy and independent, and the sum $X^{(1)} + X^{(2)} + X^{(3)}$ is the desired process.
For $X^{(1)}$ I can see why its Levy, also computing its characteristic exponent is easy and yields $i \langle b, \theta \rangle - \frac{1}{2} \langle \theta , M \theta \rangle$. But now I'm struggling with $X^{(2)}$. First off, I have no idea what the definition of $X^{(2)}$ really means. In the script, it says:
"Note that this is nothing but a random finite sum, and in fact a compound Poisson process."
I dont see why this is the case. How does the compound Poisson process precisely look like? I don't really know how I can handle the two integrals, let alone how to see that it is Levy. Is there a book where I can find more details about this specific construction? I would really like to understand it. Thanks in advance!