Laws of large numbers for dynamical systems with randomly matched individuals

Biologists and economists have analyzed populations where each individual interacts with randomly selected individuals. The random matching generates a very complicated stochastic system. Consequently biologists and economists have approximated such a system with a deterministic system. The justification for such an approximation is that the population is assumed to be very large and thus some law of large numbers must hold. This paper gives a characterization of random matching schemes for countably infinite populations. In particular this paper shows that there exists a random matching scheme such that the stochastic system and the deterministic system are the same.