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,ω )=_n=0^+∞ε_n(ω_1)ξ_n(ω_2)a_nz^n. Here (ε_n) is a sequence of independent Steinhaus random variables, (ξ_n) is a sequence of independent standard complex Gaussian random variables, and a sequence of numbers a_nC such that a_0 eq0, _n→+∞[n]|a_n|=1, {|a_n|nN}=+∞. We investigate asymptotic estimates of the probability p_0(r)=^-P{ωf(z,ω ) has no zeros inside rD} as r1 outside some set E of finite logarithmic measure. Denote N(r):=#{n|a_n|r^n>1}, s(r):=2_n=0^+∞^+(|a_n|r^n), α:=_r1N(r)11-r . The article, in particular, proves the following statements: 1) if α>4 then _r1r otin E (p_0(r)- s(r))N(r)=1; 2) if α=+∞ then 0≤_r1r otin E (p_0(r)- s(r))s(r),_r1r otin E (p_0(r)- s(r))s(r)≤12. 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 \![p. 119]Nishry2013 for such random functions. , number=1, journal=Matematychni Studii, author=Kuryliak, A. O. and Skaskiv, O. B., year=2023, month=Mar., pages=29-45