An analytic continuation of random analytic functions (in Ukrainian)
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$varlimsuplimits_{noinfty}oot{n}of{|a_n|}=1$, theci...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | German |
| Published: |
Ivan Franko National University of Lviv
2011-11-01
|
| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/texts/2011/36_2/128-132.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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$varlimsuplimits_{noinfty}oot{n}of{|a_n|}=1$, thecircle ${zinmathbb{C}colon |z|=1}$ 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$varliminflimits_{koinfty}frac{n_k}k<+infty$. |
|---|---|
| ISSN: | 1027-4634 |