  We consider the preferential attachment model. This is a growing random graph
such that at each step a new vertex is added and forms $m$ connections. The
neighbors of the new vertex are chosen at random with probability proportional
to their degree. It is well known that the proportion of nodes with a given
degree at step $n$ converges to a constant as $n\rightarrow\infty$. The goal of
this paper is to investigate the asymptotic distribution of the fluctuations
around this limiting value. We prove a central limit theorem for the joint
distribution of all degree counts. In particular, we give an explicit
expression for the asymptotic covariance. This expression is rather complex, so
we compute it numerically for various parameter choices. We also use numerical
simulations to argue that the convergence is quite fast. The proof relies on
the careful construction of an appropriate martingale.
