  We study the joint degree counts in proportional attachment random graphs and
find a simple representation for the limit distribution in infinite sequence
space. We show weak convergence with respect to the p-norm topology for
appropriate p and also provide optimal rates of convergence of the finite
dimensional distributions. The results hold for models with any general initial
seed graph and any fixed number of initial outgoing edges per vertex; we
generate non-tree graphs using both a lumping and a sequential rule.
Convergence of the order statistics and optimal rates of convergence to the
maximum of the degrees is also established.
