Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 6 / d / Solution

The joint process is a two-dimensional Compound Poisson process whose Lévy measure isTwo coordinates of a Lévy process are independent exactly when its Lévy measure charges only the coordinate axes and its Gaussian covariance has no cross term. Here there is no Gaussian part, so independence is equivalent toSince $g_1{g}_2$ is continuous, this is equivalent to pointwise vanishing. Conversely, when the product vanishes, the mark sets where $g_1$ and $g_2$ are nonzero are disjoint; independent thinning of the Poisson random measure gives independent coordinate processes. Hence