An analytic continuation of random analytic functions (in Ukrainian)

Let $(eta_n(omega))$ be a sequence of independent random variables 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 $varlimsuplimits_{ntoinfty}root{n}of{|a_n|}=1$, the circle ${zinmathbb{C}colon |z|=1}$ is the natural boundary for the function $f_omega(z)=sum_{n=0}^infty a_neta_n(omega)z^n$ almost surely if and only if the condition $sum_{k=0}^infty q_{n_k}=+infty$ holds for every increasing sequence $(n_k)$ of nonnegative integers such that $varliminflimits_{ktoinfty}frac{n_k}k