  We prove a central limit theorem for the components of the largest
eigenvectors of the adjacency matrix of a finite-dimensional random dot product
graph whose true latent positions are unknown. In particular, we follow the
methodology outlined in \citet{sussman2012universally} to construct consistent
estimates for the latent positions, and we show that the appropriately scaled
differences between the estimated and true latent positions converge to a
mixture of Gaussian random variables. As a corollary, we obtain a central limit
theorem for the first eigenvector of the adjacency matrix of an Erd\"os-Renyi
random graph.
