Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions
For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matri...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2024 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2024
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212652 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions. Armine Bagyan and Donald Richards. SIGMA 20 (2024), 094, 22 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862641551228796928 |
|---|---|
| author | Bagyan, Armine Richards, Donald |
| author_facet | Bagyan, Armine Richards, Donald |
| citation_txt | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions. Armine Bagyan and Donald Richards. SIGMA 20 (2024), 094, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on d,ₚ. The accuracy of each truncated expansion is strictly increasing in ; also, for sufficiently large , the accuracy is strictly increasing in , the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both → ∞ and → ∞. Values of and arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as or increases. These results extend recently obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.
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| first_indexed | 2026-03-15T06:55:41Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212652 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T06:55:41Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bagyan, Armine Richards, Donald 2026-02-09T09:34:00Z 2024 Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions. Armine Bagyan and Donald Richards. SIGMA 20 (2024), 094, 22 pages 1815-0659 2020 Mathematics Subject Classification: 60E05; 62H11; 62E20; 62R30 arXiv:2402.08663 https://nasplib.isofts.kiev.ua/handle/123456789/212652 https://doi.org/10.3842/SIGMA.2024.094 For positive integers and p such that ≥ , let ℝᵈˣᵖ denote the set of × real matrices, ₚ be the identity matrix of order , and d,ₚ = { ∈ ℝᵈˣᵖ ∣ ′ = ₚ} be the Stiefel manifold in ℝᵈˣᵖ. Complete asymptotic expansions as → ∞ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on d,ₚ. The accuracy of each truncated expansion is strictly increasing in ; also, for sufficiently large , the accuracy is strictly increasing in , the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both → ∞ and → ∞. Values of and arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as or increases. These results extend recently obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions. We are grateful to the referees and the editor who read our manuscript with great thoroughness and provided us with incisive and perceptive comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions Article published earlier |
| spellingShingle | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions Bagyan, Armine Richards, Donald |
| title | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_full | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_fullStr | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_full_unstemmed | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_short | Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions |
| title_sort | complete asymptotic expansions for the normalizing constants of high-dimensional matrix bingham and matrix langevin distributions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212652 |
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