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

  • A. O. Kuryliak Ivan Franko National University of Lviv, Lviv, Ukraine
  • M. R. Kuryliak Ivan Franko National University of Lviv, Lviv, Ukraine
  • O. M. Trusevych Lviv State University of Life Safety, Lviv, Ukraine
Keywords: analytic function;, Wiman’s inequality;, maximum modulus;, maximal term;, sub-gaussian random variables;, sub-exponential random variables;, dependent random variables

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}. $ 

Author Biographies

A. O. Kuryliak, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

M. R. Kuryliak, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

O. M. Trusevych, Lviv State University of Life Safety, Lviv, Ukraine

Lviv State University of Life Safety, Lviv, Ukraine

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Published
2024-09-18
How to Cite
Kuryliak, A. O., Kuryliak, M. R., & Trusevych, O. M. (2024). Arbitrary random variables and Wiman’s inequality for analytic functions in the unit disc. Matematychni Studii, 62(1), 39-45. https://doi.org/10.30970/ms.62.1.39-45
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Articles