Analytic Gaussian functions in the unit disc: probability of zeros absence - Kuryliak, A. O.; Skaskiv, O. B. (Matematychni Studii, 2023)
In the paper we consider a random analytic function of the form$$f(z,ω )=\sum\limits_{n=0}^{+∞}ε_n(ω_1)ξ_n(ω_2)a_nz^n.$$Here $(ε_n)$ is a sequence of inde\-pendent Steinhausrandom variables, $(ξ_n)$ is a sequence of indepen\-dent standard complex Gaussianrandom variables, and a sequence of numbers $a_n\in\mathbb{C}$such that$a_0 eq0,\ \varlimsup\limits_{n→+∞}\sqrt[n]{|a_n|}=1,\ \sup\{|a_n|\colon n\in\mathbb{N}\}=+∞.$We investigate asymptotic estimates of theprobability $p_0(r)=\ln^-P\{ω\colon f(z,ω )$ 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}^{+∞}\ln^+(|a_n|r^{n}),$$ α:=\varliminf\limits_{r\uparrow1}\frac{\ln N(r)}{\ln\frac{1}{1-r}}.$ The article, in particular, proves the following statements:1) if $α>4$ then$\displaystyle\lim\limits_{\stackrel{r\uparrow1}{r otin E}}\frac{\ln(p_0(r)- s(r))}{\ln N(r)}=1$;2) if $α=+∞$ then$\displaystyle 0≤\varliminf\limits_{\stackrel{r\uparrow1}{r otin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)},\quad \varlimsup\limits_{\stackrel {r\uparrow1}{r otin E}}\frac{\ln(p_0(r)- s(r))}{\ln s(r)}≤\frac1{2}.$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.