Distribution of unit mass on one fractal self-similar web-type curve
In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable $\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random vari...
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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/547 |
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| author | M. V. Pratsiovytyi I. M. Lysenko S. P. Ratushniak O. A. Tsokolenko |
| author_facet | M. V. Pratsiovytyi I. M. Lysenko S. P. Ratushniak O. A. Tsokolenko |
| author_sort | M. V. Pratsiovytyi |
| collection | DOAJ |
| description | In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable
$\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random variables taking the values $0,1,\cdots,6$ with the probabilities $p_{0n}$, $p_{1n},\cdots,p_{6n}$; $\varepsilon_{6}=0$, $\varepsilon_0$, $\varepsilon_1,\cdots,\varepsilon_5$ are 6th roots of unity.
We prove that the set of values of random variable $\tau$ is self-similar six petal snowflake which is a fractal curve $G$ of spider web type with dimension $\log_37$. Its outline is the Koch snowflake.
We establish that $\tau$ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable $\tau$, which is the tail set of the seven-symbol representation of the points of the curve $G$.
In the case of identical distribution of the random variables $\tau_n$ (namely: $p_{kn}=p_k$) we establish that the spectrum of distribution $\tau$ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension $\alpha_0(E)=\frac{\ln {p_0^{p_0}\cdots p_6^{p_6}}}{-\ln 7}$ with respect to the Hausdorff-Billingsley $\alpha$-measure. The measure is a probabilistic generalization of the Hausdorff $\alpha$-measure. In this case, the random variables $\tau=\Delta^g_{\tau_1\cdots\tau_n\cdots}$ and $\tau'=\Delta^g_{\tau_1'...\tau_n'...}$ defined by different probability vectors $(p_0,\cdots,p_6)$ and $(p'_0,\cdots,p'_6)$ have mutually orthogonal distributions. |
| format | Article |
| id | doaj-art-e3694aede2774befbc2d8c35eaa6414b |
| institution | DOAJ |
| 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-e3694aede2774befbc2d8c35eaa6414b2025-08-20T02:40:18ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621213010.30970/ms.62.1.21-30547Distribution of unit mass on one fractal self-similar web-type curveM. V. Pratsiovytyi0I. M. Lysenko1S. P. Ratushniak2O. A. Tsokolenko3Institute of Mathematics of NASU Mykhailo Drahomanov Ukrainian State University Kyiv, UkraineInstitute of Mathematics of NASU Mykhailo Drahomanov Ukrainian State University Kyiv, Ukraine Institute of Mathematics of NASU Mykhailo Drahomanov Ukrainian State University Kyiv, UkraineInstitute of Mathematics of NASU Mykhailo Drahomanov Ukrainian State University Kyiv, UkraineIn the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable $\tau=\sum\nolimits_{n=1}^{\infty}\frac{2\varepsilon_{\tau}}{3^n}\equiv\Delta^g_{\tau_1...\tau_n...}$, where $(\tau_n)$ is a~sequence of independent random variables taking the values $0,1,\cdots,6$ with the probabilities $p_{0n}$, $p_{1n},\cdots,p_{6n}$; $\varepsilon_{6}=0$, $\varepsilon_0$, $\varepsilon_1,\cdots,\varepsilon_5$ are 6th roots of unity. We prove that the set of values of random variable $\tau$ is self-similar six petal snowflake which is a fractal curve $G$ of spider web type with dimension $\log_37$. Its outline is the Koch snowflake. We establish that $\tau$ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable $\tau$, which is the tail set of the seven-symbol representation of the points of the curve $G$. In the case of identical distribution of the random variables $\tau_n$ (namely: $p_{kn}=p_k$) we establish that the spectrum of distribution $\tau$ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension $\alpha_0(E)=\frac{\ln {p_0^{p_0}\cdots p_6^{p_6}}}{-\ln 7}$ with respect to the Hausdorff-Billingsley $\alpha$-measure. The measure is a probabilistic generalization of the Hausdorff $\alpha$-measure. In this case, the random variables $\tau=\Delta^g_{\tau_1\cdots\tau_n\cdots}$ and $\tau'=\Delta^g_{\tau_1'...\tau_n'...}$ defined by different probability vectors $(p_0,\cdots,p_6)$ and $(p'_0,\cdots,p'_6)$ have mutually orthogonal distributions.http://matstud.org.ua/ojs/index.php/matstud/article/view/547self-similar figurefractal curvespectrum of distributionlebesgue purity of the distributiondiscrete distribution with dense point spectrumsingular distributionhausdorff-besicovitсh dimensionfractal besicovitch-eggleston type set |
| spellingShingle | M. V. Pratsiovytyi I. M. Lysenko S. P. Ratushniak O. A. Tsokolenko Distribution of unit mass on one fractal self-similar web-type curve Математичні Студії self-similar figure fractal curve spectrum of distribution lebesgue purity of the distribution discrete distribution with dense point spectrum singular distribution hausdorff-besicovitсh dimension fractal besicovitch-eggleston type set |
| title | Distribution of unit mass on one fractal self-similar web-type curve |
| title_full | Distribution of unit mass on one fractal self-similar web-type curve |
| title_fullStr | Distribution of unit mass on one fractal self-similar web-type curve |
| title_full_unstemmed | Distribution of unit mass on one fractal self-similar web-type curve |
| title_short | Distribution of unit mass on one fractal self-similar web-type curve |
| title_sort | distribution of unit mass on one fractal self similar web type curve |
| topic | self-similar figure fractal curve spectrum of distribution lebesgue purity of the distribution discrete distribution with dense point spectrum singular distribution hausdorff-besicovitсh dimension fractal besicovitch-eggleston type set |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/547 |
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