Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc
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}\sq...
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Ivan Franko National University of Lviv
2024-09-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/553 |
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| author | A. O. Kuryliak M. R. Kuryliak O. M. Trusevych |
| author_facet | A. O. Kuryliak M. R. Kuryliak O. M. Trusevych |
| author_sort | A. O. Kuryliak |
| collection | DOAJ |
| description | 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}.
$ |
| format | Article |
| id | doaj-art-201ec5aeea5d4690a65d205e6015a33f |
| institution | Kabale University |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2024-09-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-201ec5aeea5d4690a65d205e6015a33f2025-08-20T03:33:17ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621394510.30970/ms.62.1.39-45553Arbitrary random variables and Wiman's inequality for analytic functions in the unit discA. O. Kuryliak0M. R. Kuryliak1O. M. Trusevych2Ivan Franko National University of Lviv, Lviv, UkraineIvan Franko National University of Lviv, Lviv, UkraineLviv State University of Life Safety, Lviv, UkraineWe 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}. $http://matstud.org.ua/ojs/index.php/matstud/article/view/553analytic function;wiman’s inequality;maximum modulus;maximal term;sub-gaussian random variables;sub-exponential random variables;dependent random variables |
| spellingShingle | A. O. Kuryliak M. R. Kuryliak O. M. Trusevych Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc Математичні Студії analytic function; wiman’s inequality; maximum modulus; maximal term; sub-gaussian random variables; sub-exponential random variables; dependent random variables |
| title | Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc |
| title_full | Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc |
| title_fullStr | Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc |
| title_full_unstemmed | Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc |
| title_short | Arbitrary random variables and Wiman's inequality for analytic functions in the unit disc |
| title_sort | arbitrary random variables and wiman s inequality for analytic functions in the unit disc |
| topic | analytic function; wiman’s inequality; maximum modulus; maximal term; sub-gaussian random variables; sub-exponential random variables; dependent random variables |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/553 |
| work_keys_str_mv | AT aokuryliak arbitraryrandomvariablesandwimansinequalityforanalyticfunctionsintheunitdisc AT mrkuryliak arbitraryrandomvariablesandwimansinequalityforanalyticfunctionsintheunitdisc AT omtrusevych arbitraryrandomvariablesandwimansinequalityforanalyticfunctionsintheunitdisc |