The growth of Taylor series and random Taylor series (Q2720088)

Consider \(f(z)= \sum^\infty_{n=0} b_nz^n\) and \(f_\omega (z)= \sum^\infty_{n=0} Z_n(\omega) b_nz^n\), where \(\{b_n\} \subset\mathbb{C}\) and \(\{Z_n (\omega)\}\) is a sequence of independent and symmetric complex or real random variables of the same distribution and veryfying some conditions in a probability space \((\Omega, {\mathfrak A},P)\). Suppose that \(\varlimsup\root n \of {b_n}=0\) and \(\varlimsup\root n\of {|b_nZ_n (\omega)|}=0\) a.s. \((n\to \infty)\) and let \(M(r,f)= \max\{|f(z)|: |z|=r\}\), \(M(r, f_\omega) =\max\{|f_\omega(z) |:|z|= r\}\). The authors prove that (1) \(\varlimsup (\ln\ln M(r,f)/ \ln\ln r)= \lambda\) \((n\to\infty)\) and (2) \(\varlimsup (\ln\ln M(r, f_\omega)/ \ln\ln r)= \lambda\) a.s. \((n\to \infty)\) if and only if NEWLINE\[NEWLINE\varlimsup [\ln n/(|n|-|n|b_n|/n |)]+ 1= \lambda \in(1,\infty)\;(n\to\infty).NEWLINE\]NEWLINE They replace \(\varlimsup\) in (1) and (2) by lim and, for existence of these limits, give also some necessary and sufficient conditions which are the above condition and an additional condition. The results are analogous to the classical ones.