We study the images of the complex Ginibre eigenvalues under the power maps $\pi_M: z \mapsto z^M$, for any integer $M$. We establish the following equality in distribution, $$ {\rm{Gin}}(N)^M \stackrel{d}{=} \bigcup_{k=1}^M {\rm{Gin}} (N,M,k), $$ where the so-called Power-Ginibre distributions ${\rm{Gin}}(N,M,k)$ form $M$ independent determinantal point processes. This result can be extended to any radially symmetric normal matrix ensemble, and generalizes Rains' superposition theorem for the CUE to a wider class of point processes. In the same spirit, we prove a generalization of Kostlan's and Rains' independence theorems for two-dimensional beta ensembles with radial symmetry and even parameter $\beta$, replacing independence by conditional independence. Our proof technique also allows us to recover a result by Edelman and La Croix for the GUE, and gives new insight into some of the questions they raised. 查看全文>>