Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials
In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov triple E,μ,Γ, where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/9408651 |
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| _version_ | 1832561338390413312 |
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| author | Yoon Tae Kim Hyun Suk Park |
| author_facet | Yoon Tae Kim Hyun Suk Park |
| author_sort | Yoon Tae Kim |
| collection | DOAJ |
| description | In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov triple E,μ,Γ, where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one. |
| format | Article |
| id | doaj-art-150864b436e640959ee54a87fd821857 |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-150864b436e640959ee54a87fd8218572025-02-03T01:25:08ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/94086519408651Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal PolynomialsYoon Tae Kim0Hyun Suk Park1Department of Statistics and Data Science Convergence Research Center, Hallym University, Chuncheon, Gangwon-Do 200-702, Republic of KoreaDepartment of Statistics and Data Science Convergence Research Center, Hallym University, Chuncheon, Gangwon-Do 200-702, Republic of KoreaIn this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov triple E,μ,Γ, where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.http://dx.doi.org/10.1155/2021/9408651 |
| spellingShingle | Yoon Tae Kim Hyun Suk Park Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials Journal of Function Spaces |
| title | Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials |
| title_full | Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials |
| title_fullStr | Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials |
| title_full_unstemmed | Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials |
| title_short | Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials |
| title_sort | quantitative fourth moment theorem of functions on the markov triple and orthogonal polynomials |
| url | http://dx.doi.org/10.1155/2021/9408651 |
| work_keys_str_mv | AT yoontaekim quantitativefourthmomenttheoremoffunctionsonthemarkovtripleandorthogonalpolynomials AT hyunsukpark quantitativefourthmomenttheoremoffunctionsonthemarkovtripleandorthogonalpolynomials |