A Central Limit Theorem via Differential Equations

1. Main theorem. In this paper, we consider parameters defined on random discrete processes. When the parameters change by only a small amount from one state in the process to the next, one often finds that the parameters satisfy a law of large numbers, that is, the parameters are sharply concentrated around certain values. Wormald [6] gives some general criteria which ensure that given parameters converge in probability to the solution of a system of differential equations. In fact, such parameters often satisfy not only a law of large numbers, but also a central limit theorem. Based on the differential equation method described in [6] and a martingale central limit theorem due to McLeish [3], we show that when certain general criteria are satisfied, a set of parameters defined on a family of discrete random processes converges to a multivariate normal distribution. As examples of processes to which this method can be applied, we consider in Sections 4 and 5 two random graph processes. In both processes, the initial state is an empty graph on n vertices, and edges are added one by one according to a random procedure. Consider a sequence (Ωn,Fn, Pn) of probability spaces. Let mn be a sequence of numbers such thatmn =O(n), and suppose that for each n a filtration Fn,0 ⊆ Fn,1 ⊆ · · · ⊆ Fn,mn ⊆ Fn is given. Let {Xn,m;m = 0,1, . . . ,mn} be a sequence of random vectors in Rq, for some q ≥ 1, such that Xn,m is