The Size of a Political Club

We investigate the size of a political club that admits new members according to a rule that if only a few members vote against admission the candidate joins, but the members who voted against admission leave. For the simplest ergodic case, we use a modified Bessel function to give the generating function for the invariant distribution of the club size. This also allows us to give asymptotics for the moments as the probability of voting for admission approaches 1 and to prove global and local central limit theorems. We also calculate the mean recurrent time and determine formulas for calculating first passage times. We then show that for an expanded set of cases the club size process approaches the Ornstein-Uhlenbeck process as the probability of voting for admission approaches 1. This means that the club size will approach a Gaussian distribution for a wide set of formal and informal admission rules. Finally, we show a comparison between theoretical results and simulation runs for two cases.