How likely is an i.i.d. degree sequence to be graphical?

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.