Analytic Gaussian functions in the unit disc: probability of zeros absence

  • A. O. Kuryliak Ivan Franko National University of Lviv, Lviv, Ukraine
  • O. B. Skaskiv Department of Mechanics and Mathematics Ivan Franko National University of Lviv Lviv, Ukraine
Keywords: Gaussian analytic functions; Steinhaus analytic functions, zeros distribution of random analytic functions.

Abstract

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.

Author Biographies

A. O. Kuryliak, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

O. B. Skaskiv, Department of Mechanics and Mathematics Ivan Franko National University of Lviv Lviv, Ukraine

Department of Mechanics and Mathematics
Ivan Franko National University of Lviv
Lviv, Ukraine

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Published
2023-03-28
How to Cite
Kuryliak, A. O., & Skaskiv, O. B. (2023). Analytic Gaussian functions in the unit disc: probability of zeros absence. Matematychni Studii, 59(1), 29-45. https://doi.org/10.30970/ms.59.1.29-45
Section
Articles