If $$\vec x \sim \mbox{dirichlet}(\alpha \mathbf1_k ),$$
what is the distribution (or approximation of): $$\sum_{i=1}^k \log(x_i)$$.
I was able to find the solution using the characteristic function following [1] for: $$ x_i \sim \mbox{gamma}(\alpha,1),$$
Which I thought was close, but adding the simplex constraint of the dirichlet distribution seems to make it more difficult...
[1] Marques, F. J., Coelho, C. A., & de Carvalho, M. (2014). On the distribution of linear combinations of independent Gumbel random variables. Statistics and Computing, 25(3), 683–701. http://doi.org/10.1007/s11222-014-9453-5