  Let $\xi_0,\xi_1,\ldots$ be independent identically distributed complex-
valued random variables such that $\mathbb{E}\log(1+|\xi _0|)<\infty$. We
consider random analytic functions of the form
\[\mathbf{G}_n(z)=\sum_{k=0}^{\infty}\xi_kf_{k,n}z^k,\] where $f_{k,n}$ are
deterministic complex coefficients. Let $\mu_n$ be the random measure counting
the complex zeros of $\mathbf{G}_n$ according to their multiplicities. Assuming
essentially that $-\frac{1}{n}\log f_{[tn],n}\to u(t)$ as $n\to\infty$, where
$u(t)$ is some function, we show that the measure $\frac{1}{n}\mu_n$ converges
in probability to some deterministic measure $\mu$ which is characterized in
terms of the Legendre-Fenchel transform of $u$. The limiting measure $\mu$ does
not depend on the distribution of the $\xi_k$'s. This result is applied to
several ensembles of random analytic functions including the ensembles
corresponding to the three two-dimensional geometries of constant curvature. As
another application, we prove a random polynomial analogue of the circular law
for random matrices.
