Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc

Abstract

We consider the class $\mathcal{A}(\varphi,\beta)$ of random analytic functions in the unit disk $\mathbb{C}=\{z\colon |z|<1\}$ of the form $f(z,\omega)=f(z,\omega_1,\omega_2)=\sum_{n=0}^{+\infty} R_n(\omega_1)\xi_n(\omega_2)a_nz^n,$ where $a_n\in\mathbb{C}\colon \lim\limits_{n\to+\infty}\sqrt[n]{|a_n|}=1,$ $\big(R_n(\omega)\big)$ is the Rademacher sequence, $\big(\xi_n(\omega)\big)$ is a sequence of complex-valued random variables (denote by $\Delta_{\varphi}$) such that there exists a constant $\beta>0$ and a function $\varphi(N,\beta)\colon\mathbb{N}\times\mathbb{R}_+\to[1;+\infty)$ non-decreasing by $N$ and $\beta$ for which $\displaystyle \Bigl(\mathbf{E}\Bigl(\max_{0\leq n\leq N}|\xi_n|^{\beta}\Bigl)\Bigl)^{1/\beta}\asymp\varphi(N,\beta),\ \ N\to+\infty,\ \ \alpha=\varlimsup_{N\to+\infty}\frac{\ln\varphi(N,\beta)}{\ln N}<+\infty,$ $\displaystyle (\exists \gamma>0)(\exists n_0\in\mathbb{N})\colon \sup\{ \mathbf{E}|\xi_n|^{-\gamma}\colon\ n\geq n_0\}<+\infty.$ By $\mathcal{A}_1(\varphi,\beta)$ we denote the class of random analytic functions in $\mathbb{D}$ of the form $f(z,\omega)=\sum_{n=0}^{+\infty} \xi_n(\omega)a_nz^n,$ where a sequence $\big(\xi_n(\omega)\big)\in\Delta_\varphi$ and, in particular, may be not sub-gaussian and not independent. In the paper, there are proved the following statements: Let $\delta>0.$ 1) Theorem 3: For $f\in\mathcal{A}(\varphi,\beta)$ there exist $r_0(\omega)>0$, a set $E(\delta)\subset(0;1)$ of finite logarithmic measure such that for all $r\in(r_0(\omega);1)\backslash E$ we have with probability $p\in(0;1)$ $\displaystyle M_f(r,\omega) \leq\frac{\mu_f(r)}{(1-p)^{1/\beta}}\varphi(N(r),\beta)\Big((1-r)^{-2}\cdot \ln\frac{\mu_f(r)\varphi(N(r),\beta)}{(1-p)(1-r)}\Big)^{1/4+\delta}. $ 2) Theorem 4: For a function $f\in\mathcal{A}_1(\varphi,\beta)$ there exist $r_0(\omega)>0$, a set $E(\delta)\subset(0;1)$ of finite logarithmic measure such that for all $r\in(r_0(\omega);1)\backslash E$ we get with probability $p\in(0;1)$ $\displaystyle M_f(r,\omega) \leq\frac{\mu_f(r)}{(1-p)^{1/\beta}}\varphi(N(r),\beta)\Big((1-r)^{-2}\cdot \ln\frac{\mu_f(r)\varphi(N(r),\beta)}{(1-p)(1-r)}\Big)^{1/2+\delta}. $