Gaussian Analytic functions in the unit ball

We study some properties of hyperbolic Gaussian analytic functions of intensity $L$ in the unit ball of ${\mathbb{C}}^n$. First we deal with the asymptotics of fluctuations of linear statistics as $L\to \mathrm{\infty }$. Then we estimate the probability of large deviations (with respect to the expected value) of such linear statistics and use this estimate to prove a hole theorem.