  Preferential attachment is widely used to model power-law behavior of degree
distributions in both directed and undirected networks. In a directed
preferential attachment model, despite the well-known marginal power-law degree
distributions, not much investigation has been done on the joint behavior of
the in- and out-degree growth. Also, statistical estimates of the marginal tail
exponent of the power-law degree distribution often use the Hill estimator as
one of the key summary statistics, even though no theoretical justification has
been given. This paper focuses on convergence of the joint empirical measure
for in- and out-degrees and proves the consistency of the Hill estimator. To do
this, we first derive the asymptotic behavior of the joint degree sequences by
embedding the in- and out-degrees of a fixed node into a pair of switched birth
processes with immigration and then establish the convergence of the joint tail
empirical measure. From these steps, the consistency of the Hill estimators is
obtained.
