Poisson convergence and random graph

1. Introduction Approximation by the Poisson distribution arises naturally in the theory of random graphs, as in many other fields, when counting the number of occurrences of individually rare and unrelated events within a large ensemble. For example, one may be concerned with the number of times that a particular small configuration is repeated in a large graph, such questions being considered, amongst others, in the fundamental paper of Erdo's and R6nyi(4). The technique normally used to obtain such approximations in random graph theory is based on showing that the factorial moments of the quantity concerned converge to those of a Poisson distribution as the size of the graph tends to infinity. Since the rth factorial moment is just the expected number of ordered r-tuples of events occurring, it is particularly well suited to evaluation by combinatorial methods. Unfortunately, such a technique becomes very difficult to manage if the mean of the approximating Poisson distribution is itself increasing with the size of the graph, and this limits the scope of the results obtainable. The purpose of this paper is to show how an alternative approach, based on that of Stein (6) for the Normal approximation, and developed in the Poisson context by Chen(3) and Barbour and Eagleson(i), can be adapted to circumvent this particular difficulty. The new approach is doubly attractive in that, at the same time, it yields an upper bound on the total variation distance between the approximating and the true distributions. The technique is described in Section 2, in the context of some classical problems raised by Erdos and Re"nyi. In Section 3, Stein's method is used to establish convergence in distribution to the Normal, together with the correct rate of convergence, in a case where a Poisson approximation would not be appropriate.