The distribution of the sum of $n$ random coplanar unit vectors and of a given component of the sum has been discussed by many authors, who have shown that each distribution can be approximated in series that are asymptotically normal. But the difficult question of the usefulness of these approximations for finite $n$--in particular for small $n$--has not been exhaustively treated. Accordingly, this paper reexamines some analyses of Pearson's series for the vector sum, presents corresponding series for a component, and examines the accuracy of the latter series.
David Durand. J. Arthur Greenwood. "Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions." Ann. Math. Statist. 28 (4) 978 - 986, December, 1957. https://doi.org/10.1214/aoms/1177706798