Independent coordinates of a compound Poisson process
= Independent coordinates of a compound Poisson process
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 to
$$
g_1(y)g_2(y)=0
$$
for every $y\in[0,1]$.