An analytic continuation of random analytic functions (in Ukrainian) - P. V. Filevych (Математичні Студії, 2011)
Let $(eta_n(omega))$ be a sequence of independent randomvariables such that $eta_n(omega)$ takes the values $-1$ and$1$ with the probabilities $p_n$ and $1-p_n$, respectively. Put$q_n=min{p_n,1-p_n}$. Then, for each complex sequence$(a_n)$ such that$varlimsuplimits_{noinfty}ootnof{|a_n|}=1$, thecircle ${zinmathbbCcolon |z|=1}$ is the natural boundaryfor the function $f_omega(z)=sum_{n=0}^inftya_neta_n(omega)z^n$ almost surely if and only if thecondition $sum_{k=0}^infty q_{n_k}=+infty$ holds for everyincreasing sequence $(n_k)$ of nonnegative integers such that$varliminflimits_{koinfty}frac{n_k}k<+infty$.