Monotonicity, asymptotic normality and vertex degrees in random graphs

We exploit a result by Nerman 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 also give some simple applications to random allocations and to spacings. Finally, inspired by these results, but logically independent of them, we investigate whether a one-sided version of the Cram'{e}r–Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it.