The value distribution of random analytic functions on the unit disk

Abstract

We study the value distributions of the random analytic functions on the unit disk of the form

$$f_\omega(z)= \sum _{j=0}^{\infty}\chi_j(\omega) a_j z^j,$$

where \(a_j\in\mathbb{C}\) and \(\chi_j(\omega)\) are independent and identically distributed random variables defined on a probability space \((\Omega, \mathcal{F}, \mu)\). Some of the theorems complement the work in [6], which deals with random entire functions. We first define a family of random analytic functions in the above form, which includes Gaussian, Rademacher, and Steinhaus analytic functions. Then we prove the relationship between the integrated counting function \(N(r, a, f_\omega)\) and the \(L_2\) norm of \(f\) on the circle \(|z|=r\) as \(r\) is close to \(1\). As a by-product, we obtain Nevanlinna's second main theorem on the unit disk. Finally, we show theorems on the maximum modulus of \(f\) and \(f_\omega\) on the unit disk.