For random threshold graphs under a general fitness scaling, the empirical fraction of nodes with degree d converges only in distribution to a non-degenerate random variable, not in probability to the nodal degree pmf.
Billingsley, Convergence of Probability Measures , John Wiley & Sons, New York (NY), 1968
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Asymptotic degree distributions in random threshold graphs
For random threshold graphs under a general fitness scaling, the empirical fraction of nodes with degree d converges only in distribution to a non-degenerate random variable, not in probability to the nodal degree pmf.