On Another Type of Convergence for Intuitionistic Fuzzy Observables
The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence....
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/1/127 |
| _version_ | 1797358494139023360 |
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| author | Katarína Čunderlíková |
| author_facet | Katarína Čunderlíková |
| author_sort | Katarína Čunderlíková |
| collection | DOAJ |
| description | The convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too. |
| first_indexed | 2024-03-08T15:02:51Z |
| format | Article |
| id | doaj.art-d6a82feb072149aa8ba4c323e0086c15 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T15:02:51Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-d6a82feb072149aa8ba4c323e0086c152024-01-10T15:03:42ZengMDPI AGMathematics2227-73902023-12-0112112710.3390/math12010127On Another Type of Convergence for Intuitionistic Fuzzy ObservablesKatarína Čunderlíková0Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, SlovakiaThe convergence theorems play an important role in the theory of probability and statistics and in its application. In recent times, we studied three types of convergence of intuitionistic fuzzy observables, i.e., convergence in distribution, convergence in measure and almost everywhere convergence. In connection with this, some limit theorems, such as the central limit theorem, the weak law of large numbers, the Fisher–Tippet–Gnedenko theorem, the strong law of large numbers and its modification, have been proved. In 1997, B. Riečan studied an almost uniform convergence on D-posets, and he showed the connection between almost everywhere convergence in the Kolmogorov probability space and almost uniform convergence in D-posets. In 1999, M. Jurečková followed on from his research, and she proved the Egorov’s theorem for observables in MV-algebra using results from D-posets. Later, in 2017, the authors R. Bartková, B. Riečan and A. Tirpáková studied an almost uniform convergence and the Egorov’s theorem for fuzzy observables in the fuzzy quantum space. As the intuitionistic fuzzy sets introduced by K. T. Atanassov are an extension of the fuzzy sets introduced by L. Zadeh, it is interesting to study an almost uniform convergence on the family of the intuitionistic fuzzy sets. The aim of this contribution is to define an almost uniform convergence for intuitionistic fuzzy observables. We show the connection between the almost everywhere convergence and almost uniform convergence of a sequence of intuitionistic fuzzy observables, and we formulate a version of Egorov’s theorem for the case of intuitionistic fuzzy observables. We use the embedding of the intuitionistic fuzzy space into the suitable MV-algebra introduced by B. Riečan. We formulate the connection between the almost uniform convergence of functions of several intuitionistic fuzzy observables and almost uniform convergence of random variables in the Kolmogorov probability space too.https://www.mdpi.com/2227-7390/12/1/127intuitionistic fuzzy eventintuitionistic fuzzy observableintuitionistic fuzzy stateMV-algebraalmost everywhere convergencealmost uniform convergence |
| spellingShingle | Katarína Čunderlíková On Another Type of Convergence for Intuitionistic Fuzzy Observables Mathematics intuitionistic fuzzy event intuitionistic fuzzy observable intuitionistic fuzzy state MV-algebra almost everywhere convergence almost uniform convergence |
| title | On Another Type of Convergence for Intuitionistic Fuzzy Observables |
| title_full | On Another Type of Convergence for Intuitionistic Fuzzy Observables |
| title_fullStr | On Another Type of Convergence for Intuitionistic Fuzzy Observables |
| title_full_unstemmed | On Another Type of Convergence for Intuitionistic Fuzzy Observables |
| title_short | On Another Type of Convergence for Intuitionistic Fuzzy Observables |
| title_sort | on another type of convergence for intuitionistic fuzzy observables |
| topic | intuitionistic fuzzy event intuitionistic fuzzy observable intuitionistic fuzzy state MV-algebra almost everywhere convergence almost uniform convergence |
| url | https://www.mdpi.com/2227-7390/12/1/127 |
| work_keys_str_mv | AT katarinacunderlikova onanothertypeofconvergenceforintuitionisticfuzzyobservables |