A BERRY–ESSEEN BOUND WITH APPLICATIONS TO VERTEX DEGREE COUNTS IN THE ERDŐS–RÉNYI RANDOM GRAPH BY LARRY GOLDSTEIN1 University of Southern California

Applying Stein's method, an inductive technique and size bias coupling yields a Berry–Esseen theorem for normal approximation without the usual restriction that the coupling be bounded. The theorem is applied to counting the number of vertices in the Erd˝ os–Rényi random graph of a given degree. 1. Introduction. We present a new Berry–Esseen theorem for sums Y of dependent variables by combining Stein's method, size bias couplings and the in-ductive technique of Bolthausen (1984) originally developed for the combinatorial central limit theorem. We apply the theorem to asses the accuracy of the normal approximation to the distribution of the number of vertices of degree d in the classical Erd˝ os–Rényi (1959) random graph G n having n vertices connected by independent edges with common success probability depending on n and a parameter θ. Over the range of parameters considered, the theorem yields a bound that is the same up to constants as the one obtained earlier by Barbour, Karo´nski and Ruci´nski (1989) for the weaker smooth function metric (19). Stein's method [Stein (1972, 1986)] often proceeds by coupling a random variable Y of interest to a related variable Y , using, for example, the method of ex-changeable pairs, size bias couplings or zero bias couplings; for an overview see Chen, Goldstein and Shao (2010). The chief innovation here is the removal of an inconvenient restriction present in a number of results that provide Kolmogorov distance bounds using Stein's method, that the difference |Y − Y | between Y and the coupled Y be bounded almost surely by a constant. Through the use of an unbounded coupling, in Theorem 2.1 we are able to extend the previous work by Kordecki (1990) on the number of isolated, or degree zero, vertices of G n to all positive degrees. To describe Theorem 1.1, our general result, recall that for a nonnegative random variable Y with finite, nonzero mean μ, we say that Y s has the Y-size bias distribution if