> [!definition] > > Let $\mu \in \mathcal I(\real^d)$ be an [[Infinitely Divisible Distribution|infinitely divisible law]] associated with the [[Lévy System]] $(m, C, M)$. Fix a [[Probability|probability]] space $(\Omega, \cf, \bp)$. Let > - $\bracs{B_t: t \ge 0}$ be the standard [[Brownian Motion]] on $(\Omega, \cf, \bp)$. > - $\bracs{Z_t: t \ge 0}$ be a [[M2 Compound Poisson Process|compound Poisson process]] associated with $M$ on $(\omega, \cf, \bp)$. > > such that the two processes are independent. Let $X_t = mt + C^{1/2}B_t + Z_t$, then $X_t$ is a [[Lévy Process]] associated with $\pi_{m, C, M}$. From here, denote $X_t^G = mt + C^{1/2}B_t$ as the Gaussian component, and $X_t^P = Z_t$ as the Poisson component. > [!theorem] > > - Let $j(t, dy, X)$ be the [[Jump Measure|jump measure]] corresponding to $X$, then for any $t > 0$, $j(t, dy, X) = j(t, dy, X^P)$. > - $X$ is continuous if and only if $X^P = 0$. > - $\norm{X}_{\text{var}, [0, t]} < \infty$ for any $t > 0$ if and only if $C = 0$ and $M \in \mathfrak{M}_1$.