Tails of explosive birth processes and applications to non-linear P \'{o}lya urns

Abstract

We derive a simple expression for the tail-asymptotics of an explosive birth process at a fixed observation time conditioned on non-explosion. Using the well-established exponential embedding, we apply this result to compute the tail distribution of the number of balls of a losing colour in generalized P \'{o}lya urn models with super-linear feedback, which are known to exhibit a strong monopoly for the winning colour where losers only win a finite amount. Previous results in this direction were restricted to two colours with the same feedback, which we extend to an arbitrary finite number of colours with individual feedback mechanisms. As an apparent paradox, losing colours with weak feedback are more likely to win in many steps than those with strong feedback. Our approach also allows to characterize the correlations of several losing colours, which provides new insight in the distribution of other related quantities like the total number of balls for losing colours or the time to monopoly. In order to give a complete picture, we also consider sub-linear feedback and discuss asymptotics for a diverging number of colours.