On Hilbert C*-module-valued Random Variables
In this paper random variables that take their values from a Hilbert <i>C</i>*-module are defined and three definitions for the mean, covariance operator, and Gaussian distribution of these random variables are given and it is shown that these definitions are equivalent. Furthermore, the...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Springer
2015-11-01
|
| Series: | Journal of Statistical Theory and Applications (JSTA) |
| Subjects: | |
| Online Access: | https://www.atlantis-press.com/article/25845141.pdf |
| _version_ | 1818013438154113024 |
|---|---|
| author | K. Shafie |
| author_facet | K. Shafie |
| author_sort | K. Shafie |
| collection | DOAJ |
| description | In this paper random variables that take their values from a Hilbert <i>C</i>*-module are defined and three definitions for the mean, covariance operator, and Gaussian distribution of these random variables are given and it is shown that these definitions are equivalent. Furthermore, the concept of covariance of two real valued random variables and its properties are extended to two Hilbert <i>C</i>*-module valued random variables. These lead us to the generalization of Rao-Blackwell theorem for this type of random variables. Finally, in a special case, it is proved that the finiteness of second moment of the norm of such a random variable is a sufficient condition for the central limit theorem to be true. |
| first_indexed | 2024-04-14T06:33:09Z |
| format | Article |
| id | doaj.art-0332c2a839524660a5b3696b9b5bb599 |
| institution | Directory Open Access Journal |
| issn | 2214-1766 |
| language | English |
| last_indexed | 2024-04-14T06:33:09Z |
| publishDate | 2015-11-01 |
| publisher | Springer |
| record_format | Article |
| series | Journal of Statistical Theory and Applications (JSTA) |
| spelling | doaj.art-0332c2a839524660a5b3696b9b5bb5992022-12-22T02:07:33ZengSpringerJournal of Statistical Theory and Applications (JSTA)2214-17662015-11-0114410.2991/jsta.2015.14.4.2On Hilbert C*-module-valued Random VariablesK. ShafieIn this paper random variables that take their values from a Hilbert <i>C</i>*-module are defined and three definitions for the mean, covariance operator, and Gaussian distribution of these random variables are given and it is shown that these definitions are equivalent. Furthermore, the concept of covariance of two real valued random variables and its properties are extended to two Hilbert <i>C</i>*-module valued random variables. These lead us to the generalization of Rao-Blackwell theorem for this type of random variables. Finally, in a special case, it is proved that the finiteness of second moment of the norm of such a random variable is a sufficient condition for the central limit theorem to be true.https://www.atlantis-press.com/article/25845141.pdfBanach Valued random variable; central limit theoremCovariance operatorHilbert <i>C</i>*-modules |
| spellingShingle | K. Shafie On Hilbert C*-module-valued Random Variables Journal of Statistical Theory and Applications (JSTA) Banach Valued random variable; central limit theorem Covariance operator Hilbert <i>C</i>*-modules |
| title | On Hilbert C*-module-valued Random Variables |
| title_full | On Hilbert C*-module-valued Random Variables |
| title_fullStr | On Hilbert C*-module-valued Random Variables |
| title_full_unstemmed | On Hilbert C*-module-valued Random Variables |
| title_short | On Hilbert C*-module-valued Random Variables |
| title_sort | on hilbert c module valued random variables |
| topic | Banach Valued random variable; central limit theorem Covariance operator Hilbert <i>C</i>*-modules |
| url | https://www.atlantis-press.com/article/25845141.pdf |
| work_keys_str_mv | AT kshafie onhilbertcmodulevaluedrandomvariables |