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.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6103 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm