An analytic continuation of random analytic functions (in Ukrainian)

Author P. V. Filevych
Львiвський нацiональний унiверситет ветеринарної медицини та бiотехнологiй iм. С. З. Ґжицького

Abstract 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 $\varlimsup\limits_{n\to\infty}\root{n}\of{|a_n|}=1$, the circle $\{z\in\mathbb{C}\colon |z|=1\}$ is the natural boundary for the function $f_\omega(z)=\sum_{n=0}^\infty a_n\eta_n(\omega)z^n$ 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 $\varliminf\limits_{k\to\infty}\frac{n_k}k<+\infty$.
Keywords random analytic function; singular point; analytic continuation; natural boundary
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Pages 128-132
Volume 36
Issue 2
Year 2011
Journal Matematychni Studii
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