Order of growth for random Taylor series (Q2714887)

Let \(f_\omega(z)= \sum^{+ \infty}_{n=0} a_nX_n (\omega)z^n\), where \(\varlimsup \root n\of {|a_n |}=0\), \(\varlimsup (\ln|a_n |)/n \ln n=-1/ \rho\) \((n\to+ \infty)\), \(\{X_n(\omega)\}\) is a sequence of independent random complex variables in a probability space \((\Omega,{\mathcal A},P)\) (\(\omega\in \Omega\)), \(\forall n\in \mathbb{N}\), \(E(X_n)=0\), \(E(|X_n |^2) =\delta^2_n >0\), \(E(|X_n|/ \delta_n)= d_n>d\) (fixed) \(>0\) and \((a,b) \subset (0,+ \infty)\) such that \(\forall n\), \(\delta_n \in[a,b]\). Then \(f_\omega (z)\) is almost surely of order \(\rho\) in the whole complex plane, on every ray from the origin and in every angular domain with vertex at the origin. The above results can be proved by a method of J.-P. Kahane in-stead of the Paley-Zygmund inequality as \textit{F.-J. Tian} and \textit{D. Sun}, and \textit{J. Yu} [C. R. Acad. Sci. Paris, Sér. I, Math. 326, No. 4, 427-431 (1998; Zbl 0920.30005)].