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

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Given i.i.d. positive integer valued random variables D₁,...,Dₙ, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D₁,...,Dₙ. We give sufficient conditions on the distribution of Dᵢ 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(Dᵢ≥ n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.