Monotonicity, asymptotic normality and vertex degrees in random graphs

We exploit a result by Nerman [23] which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n, m) with a fixed number of edges from the corresponding result for the random graph G(n, p) with independent edges. We give also some simple applications to random allocations and to spacings. Finally, inspired by these results but logically independent from them, we investigate whether a one-sided version of the Cramér–Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.