The 'Devil's Horns' Problem of Inverting Confluent Characteristic Functions

We warn of a class of problems that occur when inverting some Characteristic Functions (CFs) which we term confluent. Confluences can arise in joint CFs during the derivation of the marginal density of some lower-dimensional combination of the components. We illustrate with a simple (yet common) autoregressive model. We show that both numerical and analytical inversions of the currently-available CFs can lead to inaccuracies in the form of a sharp spike on each tail of the statistics' densities. These twin spikes are akin to devil's horns