Compound Poisson process

A compound Poisson process has the formwhere $N$ is a Poisson process and the independent identically distributed marks $Y_n$ are independent of $N$. It is a Lévy process with a finite Lévy measure.

A compound Poisson process with rate $\lambda$ and mark $Y$ is a martingale exactly when $\mathbb{E}|Y|<\infty$ and $\mathbb{E}Y=0$.

The coordinates of a vector-valued Compound Poisson process are independent exactly when its Lévy measure is supported on the union of the coordinate axes. For marks $(g_1(U),g_2(U))$ with $U$ uniform on $[0,1]$ and continuous $g_i$, this is equivalent tofor every $y\in [0,1]$.