Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction
The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and...
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| Date: | 1996 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
1996
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/5252 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511471714172928 |
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| author | Pratsiovytyi, M. V. Працьовитий, М. В. |
| author_facet | Pratsiovytyi, M. V. Працьовитий, М. В. |
| author_sort | Pratsiovytyi, M. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:28:48Z |
| description | The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent. |
| first_indexed | 2026-03-24T03:13:25Z |
| format | Article |
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| id | umjimathkievua-article-5252 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:13:25Z |
| publishDate | 1996 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e1/740528aa8773d4d2d793d2b64132c6e1.pdf |
| spelling | umjimathkievua-article-52522020-03-18T21:28:48Z Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction Сингулярність розподілів випадкових величин, заданих розподілами елементів свого ланцюгового зображення Pratsiovytyi, M. V. Працьовитий, М. В. The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent. Повністю вивчена структура розподілу випадкової величини, елементи елементарного ланцюгового зображення якої є незалежними випадковими величинами. Доведено чистоту розподілу, знайдено критерій сингулярності і доведено неможливість абсолютної неперервності, вивчено властивості спектра. Для розподілу випадкової величини, елементи ланцюгового зображення якої утворюють однорідний ланцюг Маркова, описано спектр, знайдено вираз для функції розподілу, виведено формулу для щільності, знайдено критерій канторовості і доведено відсутність абсолютно неперервної компоненти. Institute of Mathematics, NAS of Ukraine 1996-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5252 Ukrains’kyi Matematychnyi Zhurnal; Vol. 48 No. 8 (1996); 1086-1095 Український математичний журнал; Том 48 № 8 (1996); 1086-1095 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5252/7197 https://umj.imath.kiev.ua/index.php/umj/article/view/5252/7198 Copyright (c) 1996 Pratsiovytyi M. V. |
| spellingShingle | Pratsiovytyi, M. V. Працьовитий, М. В. Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title | Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title_alt | Сингулярність розподілів випадкових величин, заданих розподілами елементів свого ланцюгового зображення |
| title_full | Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title_fullStr | Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title_full_unstemmed | Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title_short | Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| title_sort | singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5252 |
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