As abstract representations of relational data, graphs and networks find wide use in a variety of fields, particularly when working in non-Euclidean spaces. Yet for graphs to be truly useful in in the context of signal processing, one ultimately must have access to flexible and tractable statistical models. One model currently in use is the Chung-Lu random graph model, in which edge probabilities are expressed in terms of a given expected degree sequence. An advantage of this model is that its parameters can be obtained via a simple, standard estimator. Although this estimator is used frequently, its statistical properties have not been fully studied. In this paper, we develop a central limit theory for a simplified version of the Chung-Lu parameter estimator. We then derive approximations for moments of the general estimator using the delta method, and confirm the effectiveness of these approximations through empirical examples.