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We provide optimal rates of convergence to the asymptotic distribution of the(properly scaled) degree of a fixed vertex in two preferential attachmentrandom graph models. Our approach is to show that these distributions areunique fixed points of certain distributional transformations which allows usto obtain rates of convergence using a new variation of Stein's method. Despitethe large literature on these models, there is surprisingly little known aboutthe limiting distributions so we also provide some properties and newrepresentations, including an explicit expression for the densities in terms ofthe confluent hypergeometric function of the second kind.

Degree asymptotics with rates for preferential attachment random graphs