  Given i.i.d. positive integer valued random variables D_1,...,D_n, one can
ask whether there is a simple graph on n vertices so that the degrees of the
vertices are D_1,...,D_n. We give sufficient conditions on the distribution of
D_i for the probability that this be the case to be asymptotically 0, {1/2} or
strictly between 0 and {1/2}. These conditions roughly correspond to whether
the limit of nP(D_i\geq n) is infinite, zero or strictly positive and finite.
This paper is motivated by the problem of modeling large communications
networks by random graphs.
