Distributions in a class of Poissonized urns with an application to Apollonian networks

We study a class of Pólya processes that underlie terminal nodes in a random Apollonian network. We calculate the exact first and second moments of the number of terminal nodes by solving ordinary differential equations. These equations are derived from the partial differential equation governing the process. In fact, the partial differential equation yields a stochastic hierarchy of moment equations, which can be bootstrapped to get higher moments from the equations that have been solved for lower moments. We also show that the number of terminal nodes, when appropriately scaled, converges in distribution to a gamma random variable via the method of moments. The asymptotic results can be obtained using classic methods of branching processes. The manuscript explores the potential of an alternative method capable of producing exact moments and rates of convergence.