1 results
A characterization of random analytic functions satisfying Blaschke-type conditions
- Part of
-
- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 17 January 2024, pp. 670-679
- Print publication:
- September 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
Let
$f(z)=\sum _{n=0}^{\infty }a_n z^n \in H(\mathbb {D})$ be an analytic function over the unit disk in the complex plane, and let 
$\mathcal {R} f$ be its randomization: 
$$ \begin{align*}(\mathcal{R} f)(z)= \sum_{n=0}^{\infty} a_n X_n z^n \in H(\mathbb{D}),\end{align*} $$where
$(X_n)_{n\ge 0}$ is a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this note, we characterize those 
$f(z) \in H(\mathbb {D})$ such that the zero set of 
$\mathcal {R} f$ satisfies a Blaschke-type condition almost surely: 
$$ \begin{align*}\sum_{n=1}^{\infty}(1-|z_n|)^t<\infty, \quad t>1.\end{align*} $$
