= Independent coordinates of a compound Poisson process
For a vector-valued <Compound Poisson process>, its coordinate processes are <independent> exactly when its <Lévy measure>[Lévy measure] is supported on the union of the <coordinate axis>[coordinate axes]. A jump can then change only one coordinate. Splitting the marked <Poisson process> by the axis of its mark gives <independent> coordinate jump processes. Conversely, <independent> finite-activity coordinates have no simultaneous nonzero jumps <almost surely>, excluding Lévy mass away from the axes.
For a positive rate and two-dimensional marks $(g_1(U),g_2(U))$, with $U$ uniform on $[0,1]$, the condition is $g_1(U)g_2(U)=0$ <almost surely>. If both <functions> are continuous, it is equivalent to
$$
g_1(u)g_2(u)=0\qquad\text{for every }u\in[0,1].
$$
A nonzero value of the continuous product would persist on an <interval> of positive uniform <probability>. Without <continuity> only the almost-everywhere condition is necessary.
For example, marks $((1/2-U)^+,(U-1/2)^+)$ give <independent> coordinates by thinning into the two half-intervals. Marks $(U,1-U)$ produce common jumps in both coordinates <almost surely>, so the coordinates are dependent even though a single scalar uniform generates the marks. The positive-rate qualification is needed: a zero-rate process has <independent> constant coordinates irrespective of its unused mark <functions>.
Back to article page
