  We consider random analytic functions defined on the unit disk of the complex
plane as power series such that the coefficients are i.i.d., complex valued
random variables, with mean zero and unit variance. For the case of complex
Gaussian coefficients, Peres and Vir\'ag showed that the zero set forms a
determinantal point process with the Bergman kernel. We show that for general
choices of random coefficients, the zero set is asymptotically given by the
same distribution near the boundary of the disk, which expresses a universality
property. The proof is elementary and general.
