Sub-Gaussian random variables and Wiman's inequality for analytic functions

Abstract

Let $f$ be an analytic function in $\{z: |z|<R\}$ of the form $f(z)=\sum\limits_{n=0}^{+\infty}a_n z^n$. In the paper, we consider the Wiman-type inequality for random analytic functions of the form $f(z,\omega)=\sum\limits_{n=0}^{+\infty}Z_n(\omega)a_nz^n$, where $(Z_n)$ is a sequence on the Steinhaus probability space of real independent centered sub-Gaussian random variables, i.e. $(\exists D>0)(\forall k\in\mathbb{N})(\forall \lambda\in\mathbb{R})\colon \mathbf{E}(e^{\lambda Z_k})\leq e^{D \lambda^2}$, and such that $(\exists\beta>0)(\exists n_0\in\mathbb{N})\colon \inf\limits_{n\geq n_0}\mathbf{E}|Z_n|^{-\beta}<+\infty.$
 It is proved that for every $\delta>0$ there exists a set $E(\delta)\subset [0,R)$ of finite $h$-logarithmic measure (i.e. $\int\nolimits_{E}h(r)d\ln r<+\infty$) such that almost surely for all $r\in(r_0(\omega),R)\backslash E$ we have \[ M_f(r,\omega):=\max\big\{|f(z,\omega)|\colon |z|=r\big\}\leq \sqrt{h(r)}\mu_f(r)\Big(\ln^3h(r)\ln\{h(r)\mu_f(r)\}\Big)^{1/4+\delta}, \] where $h(r)$ is any fixed continuous non-decreasing function on $[0;R)$ such that $h(r)\geq2$ for all $r\in (0,R)$ and $\int^R_{r_{0}} h(r) d\ln r =+\infty$ for some $r_0\in(0,R)$.