A characterization of random analytic functions satisfying Blaschke-type conditions

Let f(z) = ∑^∞_n=₀ anzⁿ ∈ H(D) be an analytic function over the unit disk in the complex plane, and let R f be its randomization: ∞ (R f)(z) = ∑ anXnzⁿ ∈ H(D), n=0 where (Xn)n≥0 is a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this note, we characterize those f(z) ∈ H(D) such that the zero set of R f satisfies a Blaschke-type condition almost surely: ∞ ∑(1 − ∣zn∣)^t < ∞, t > 1. n=1.