Informally, the results of "Asymptotic distribution of coordinates on high-dimensional spheres" by M. C. Spruill (http://www.emis.ams.org/journals/EJP-ECP/_ejpecp/ECP/include...) is that as n gets large, the distribution of x approaches a normal distribution with mean 0 and standard deviation 1/sqrt(n). That paper contains what might be some interesting references to the literature.