Superposition of many independent spike trains is generally not a Poisson process

We study the sum of many independent spike trains and ask whether the resulting spike train has Poisson statistics or not. It is shown that for a non-Poissonian statistics of the single spike train, the resulting sum of spikes has exponential interspike interval (ISI) distributions, vanishing the ISI correlation at a finite lag but exhibits exactly the same power spectrum as the original spike train does. This paradox is resolved by considering what happens to ISI correlations in the limit of an infinite number of superposed trains. Implications of our findings for stochastic models in the neurosciences are briefly discussed.