  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.
