Analytic Gaussian functions in the unit disc: probability of zeros absence
In the paper we consider a random analytic function of the form $$f(z,\omega )=\sum\limits_{n=0}^{+\infty}\varepsilon_n(\omega_1)\xi_n(\omega_2)a_nz^n.$$ Here $(\varepsilon_n)$ is a sequence of inde\-pendent Steinhaus random variables, $(\xi_n)$ is a sequence of indepen\-dent standard complex Gaussi...
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| Format: | Article |
| Language: | German |
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Ivan Franko National University of Lviv
2023-03-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/401 |
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| author | A. O. Kuryliak O. B. Skaskiv |
| author_facet | A. O. Kuryliak O. B. Skaskiv |
| author_sort | A. O. Kuryliak |
| collection | DOAJ |
| description | In the paper we consider a random analytic function of the form
$$f(z,\omega )=\sum\limits_{n=0}^{+\infty}\varepsilon_n(\omega_1)\xi_n(\omega_2)a_nz^n.$$
Here $(\varepsilon_n)$ is a sequence of inde\-pendent Steinhaus
random variables, $(\xi_n)$ is a sequence of indepen\-dent standard complex Gaussian
random variables, and a sequence of numbers $a_n\in\mathbb{C}$
such that
$a_0\neq0,\ \varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,\ \sup\{|a_n|\colon n\in\mathbb{N}\}=+\infty.$
We investigate asymptotic estimates of the
probability $p_0(r)=\ln^-P\{\omega\colon f(z,\omega )$ has
no zeros inside $r\mathbb{D}\}$ as $r\uparrow1$ outside some set $E$ of finite logarithmic measure. Denote
$
N(r):=\#\{n\colon |a_n|r^n>1\},$ $ s(r):=2\sum_{n=0}^{+\infty}\ln^+(|a_n|r^{n}),
$
$ \alpha:=\varliminf\limits_{r\uparrow1}\frac{\ln N(r)}{\ln\frac{1}{1-r}}.
$
The article, in particular, proves the following statements:
1) if $\alpha>4$ then
$\displaystyle\lim\limits_{\stackrel{r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln N(r)}=1$;
2) if $\alpha=+\infty$ then
$\displaystyle 0\leq\varliminf\limits_{\stackrel{r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)},\quad \varlimsup\limits_{\stackrel {r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)}\leq\frac1{2}.$
Here $E$ is a set of finite logarithmic measure. The obtained asymptotic estimates are in a certain sense best possible.
Also we give an answer to an open question from \!\cite[p. 119]{Nishry2013} for such random functions. |
| format | Article |
| id | doaj-art-2a0848b690c64e069c99ac2eefd71dba |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-03-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-2a0848b690c64e069c99ac2eefd71dba2025-07-08T09:05:54ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-03-01591294510.30970/ms.59.1.29-45401Analytic Gaussian functions in the unit disc: probability of zeros absenceA. O. Kuryliak0O. B. Skaskiv1Ivan Franko National University of Lviv, Lviv, UkraineDepartment of Mechanics and Mathematics Ivan Franko National University of Lviv Lviv, UkraineIn the paper we consider a random analytic function of the form $$f(z,\omega )=\sum\limits_{n=0}^{+\infty}\varepsilon_n(\omega_1)\xi_n(\omega_2)a_nz^n.$$ Here $(\varepsilon_n)$ is a sequence of inde\-pendent Steinhaus random variables, $(\xi_n)$ is a sequence of indepen\-dent standard complex Gaussian random variables, and a sequence of numbers $a_n\in\mathbb{C}$ such that $a_0\neq0,\ \varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,\ \sup\{|a_n|\colon n\in\mathbb{N}\}=+\infty.$ We investigate asymptotic estimates of the probability $p_0(r)=\ln^-P\{\omega\colon f(z,\omega )$ has no zeros inside $r\mathbb{D}\}$ as $r\uparrow1$ outside some set $E$ of finite logarithmic measure. Denote $ N(r):=\#\{n\colon |a_n|r^n>1\},$ $ s(r):=2\sum_{n=0}^{+\infty}\ln^+(|a_n|r^{n}), $ $ \alpha:=\varliminf\limits_{r\uparrow1}\frac{\ln N(r)}{\ln\frac{1}{1-r}}. $ The article, in particular, proves the following statements: 1) if $\alpha>4$ then $\displaystyle\lim\limits_{\stackrel{r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln N(r)}=1$; 2) if $\alpha=+\infty$ then $\displaystyle 0\leq\varliminf\limits_{\stackrel{r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)},\quad \varlimsup\limits_{\stackrel {r\uparrow1}{r\notin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)}\leq\frac1{2}.$ Here $E$ is a set of finite logarithmic measure. The obtained asymptotic estimates are in a certain sense best possible. Also we give an answer to an open question from \!\cite[p. 119]{Nishry2013} for such random functions.http://matstud.org.ua/ojs/index.php/matstud/article/view/401gaussian analytic functions; steinhaus analytic functions, zeros distribution of random analytic functions. |
| spellingShingle | A. O. Kuryliak O. B. Skaskiv Analytic Gaussian functions in the unit disc: probability of zeros absence Математичні Студії gaussian analytic functions; steinhaus analytic functions, zeros distribution of random analytic functions. |
| title | Analytic Gaussian functions in the unit disc: probability of zeros absence |
| title_full | Analytic Gaussian functions in the unit disc: probability of zeros absence |
| title_fullStr | Analytic Gaussian functions in the unit disc: probability of zeros absence |
| title_full_unstemmed | Analytic Gaussian functions in the unit disc: probability of zeros absence |
| title_short | Analytic Gaussian functions in the unit disc: probability of zeros absence |
| title_sort | analytic gaussian functions in the unit disc probability of zeros absence |
| topic | gaussian analytic functions; steinhaus analytic functions, zeros distribution of random analytic functions. |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/401 |
| work_keys_str_mv | AT aokuryliak analyticgaussianfunctionsintheunitdiscprobabilityofzerosabsence AT obskaskiv analyticgaussianfunctionsintheunitdiscprobabilityofzerosabsence |