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

Q3 Mathematics
A. Kuryliak, O. Skaskiv
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引用次数: 0

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 Steinhausrandom variables, $(\xi_n)$ is a sequence of indepen\-dent standard complex Gaussianrandom 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 theprobability $p_0(r)=\ln^-P\{\omega\colon f(z,\omega )$ hasno 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}),$$ \alpha:=\varliminf\limits_{r\uparrow1}\frac{\ln N(r)}{\ln\frac{1}{1-r}}.$ The article, in particular, proves the following statements:\noi 1) if $\alpha>4$ then\centerline{$\displaystyle \lim_{\begin{substack} {r\uparrow1 \\ r\notin E}\end{substack}}\frac{\ln(p_0(r)- s(r))}{\ln N(r)}=1$;} \noi2) if $\alpha=+\infty$ then\centerline{$\displaystyle 0\leq\varliminf_{\begin{substack} {r\uparrow1 \\ r\notin E}\end{substack}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)},\quad \varlimsup_{\begin{substack} {r\uparrow1 \\ r\notin E}\end{substack}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)}\leq\frac1{2}.$} \noiHere $E$ is a set of finite logarithmic measure. The obtained asymptotic estimates are in a certain sense best possible.Also we give an answer to an open question from \!\cite[p. 119]{Nishry2013} for such random functions.
单位圆盘上的解析高斯函数:零缺席的概率
在本文中,我们考虑形式为$$f(z,\omega)=\sum\limits_0}^{+\infty}\varepsilon_n(\omega_1)\xi_n(\ omega_2)a_nz^n的随机分析函数$$这里$(\varepsilon_n)$是独立的斯坦豪斯随机变量序列,$(\xi_n)美元是独立的标准复高斯随机变量序列和$a_n\in\mathbb{C}$的数字序列,使得$a_0\neq0,\\varlimsup\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,\\sup\{| a_n|\colon n\in\mathbb{n}\}=+\infity$我们研究了概率$p_0(r)=\ln^-p\{\omega\冒号f(z,\omega)$在$r\mathb{D}\}$内没有零的渐近估计,作为有限对数测度的某个集合$E$外的$r\uparrow1$。表示$N(r):=\#\{N\colon|a_N|r^N>1\},$$s(r):=2\sum_{N=0}^{+\infty}\ln^+(|a_N| r^{N}这篇文章特别证明了以下陈述:\noi 1)如果$\alpha>4$,那么\central{$\displaystyle\lim_{\sbegin{substack}{r\uparrow1\r\nnotin E}\end{subsack}}}\ frac{\ln(p_0(r)-s(r))}{\ln N(r)}=1$;}\noi2)如果$\alpha=+\infty$,则\central{$\displaystyle 0\leq\varliminf_{\boot{substack}{r\uparrow1\r\nnotin E}\end{subsack}}\frac{\ln(p_0(r)-s(r))}{\ln s(r)},\quad\varlimsup_ leq\frac1{2}.$}\noiHere$E$是一组有限对数测度。所获得的渐近估计在某种意义上是最佳可能的。此外,我们还回答了来自\!\引用〔p.119〕{Nishry2013}对于这样的随机函数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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