A central limit theorem for the number of isolated vertices in a preferential attachment random graph

We study the number of isolated vertices in a preferential attachment random graph introduced by Dereich and M"orters in 2009. In this graph model vertices are added over time and newly arriving vertices connect to older ones with probability proportional to a (sub-)linear function of the indegree of the older vertex at that time. Using Stein’s method and a size-bias coupling, we deduce bounds in the Wasserstein distance between the law of the properly rescaled number of isolated vertices and a standard Gaussian distribution.