An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds
Abstract
:1. Introduction
2. Preliminaries
2.1. Malliavin Calculus
2.2. Stein’s Method
3. Edgeworth Expansion
- (a)
- for ,
- (b)
- for ,
4. Optimal Berry–Esseen Bound
5. Applications
5.1. Stochastic Partial Differential Equation
5.2. Ornstein-Uhlenbeck Process
5.3. -Brownian Bridge
6. Conclusions and Future Works
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Nualart, D.; Peccati, G. Central limit theorems for sequences of multiple stochastic integrals. Ann. Prob. 2015, 33, 177–193. [Google Scholar] [CrossRef]
- Nourdin, I.; Peccati, G. Stein’s method on Wiener Chaos. Probab. Theory Relat. Fields 2009, 145, 75–118. [Google Scholar] [CrossRef] [Green Version]
- Nourdin, I. Lectures on Gaussian approximations with Malliavin calculus. In Proceedings of the Séminaire de Probabilités XLV; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 2013; Volume 2078. [Google Scholar] [CrossRef] [Green Version]
- Nourdin, I.; Peccati, G. Stein’s method meets Malliavin calculus: A short survey with new estimates. In Recent Development in Stochastic Dynamics and Stochasdtic Analysis; World Scientific Publishing: Hackensack, NJ, USA, 2010; pp. 207–236. [Google Scholar]
- Nourdin, I.; Peccati, G. Normal approximations with Malliavin calculus: From Stein’s method to universality. In Cambridge Tracts in Mathematica; Cambridge University Press: Cambridge, UK, 2012; Volume 192. [Google Scholar]
- Nualart, D. Malliavin calculus and related topics. In Probability and Its Applications, 2nd ed.; Springer: Berlin, Germany, 2006. [Google Scholar]
- Chen, L.H.Y.; Goldstein, L.; Shao, Q.-M. Normal Apprtoximation by Stein’s Method; Probability and Its Applications; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 2011. [Google Scholar]
- Stein, C. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory; University of California Press: Berkeley, CA, USA, 1972; pp. 583–602. [Google Scholar]
- Stein, C. Approximate Computation of Expectations; IMS: Hayward, CA, USA, 1986. [Google Scholar]
- Biermé, H.; Bonami, A.; Nourdin, I.; Peccati, G. Optimal Berry–Esseen rates on the Wiener space: The barrier of third and fourth cumulants. ALEA Lat. Am. J. Probab. Math. Stat. 2012, 9, 473–500. [Google Scholar]
- Nourdin, I.; Peccati, G. The optimal fourth moment theorem. Proc. Am. Math. Soc. 2015, 143, 3123–3133. [Google Scholar] [CrossRef] [Green Version]
- Kim, Y.T.; Park, H.S. Optimal Berry–Esseen bound for statistical estimations and its application to SPDE. J. Multi. Anal. 2017, 155, 284–304. [Google Scholar] [CrossRef]
- Kim, Y.T.; Park, H.S. An Edeworth expansion for functionals of Gaussian fields and its applications. Stoch. Process. Their Appl. 2018, 44, 312–320. [Google Scholar]
- Hall, P. The Bootstrap and Edgeworth Expansion; Springer Series in Statistics; Springer: New York, NY, USA, 1992. [Google Scholar]
- McCullagh, P. Tensor methods in statistics. In Monographs on Statistics and Applied Probability; Chapman & Hall: London, UK, 1987. [Google Scholar]
- Nourdin, I.; Peccati, G. Cumulants on the Wiener space. J. Funct. Anal. 2010, 258, 3775–3791. [Google Scholar] [CrossRef] [Green Version]
- Leonov, V.P.; Shiryaev, A.N. On the method of calculations of semi-invariants. Theory Probab. Its Appl. 1959, 4, 319–329. [Google Scholar] [CrossRef]
- Peccati, G.; Taqqu, M. Moments, cumulants and diagram formulae for non-linear functionals of random measures. arXiv 2008, arXiv:0811.1726. [Google Scholar]
- Es-Sebaiy, K.; Moustaaid, J. Optimal Berry-Esséen bound for Maximum likelihood estimation of the drift parameter in α-Brownian bridge. J. Korean Stat. Soc. 2021, 50, 403–418. [Google Scholar] [CrossRef]
- Kim, Y.T.; Park, H.S. Optimal Berry–Esseen bound for an estimator of parameter in the Ornstein-Uhlenbeck process. J. Korean Statist. Soc. 2017, 46, 413–425. [Google Scholar] [CrossRef]
- Huebner, M.; Rozovskii, B.L. On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s. Probab. Theory Relat. Fields 1995, 103, 143–163. [Google Scholar] [CrossRef]
- Chen, Y.; Kuang, N.; Li, Y. Berry-Esséen bound for the parameter estimation of fractional Ornstein-Uhlenbeck processes. Stochastics Dyn. 2020, 20, 2050023. [Google Scholar] [CrossRef] [Green Version]
- Es-Sebaiy, K. Berry-Esséen bounds for the least squares estimator for discretely observed fractional Ornstein-Uhlenbeck processes. Stat. Prob. Lett. 2012, 83, 2372–2385. [Google Scholar] [CrossRef] [Green Version]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Kim, Y.-T.; Park, H.-S. An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds. Mathematics 2021, 9, 2223. https://doi.org/10.3390/math9182223
Kim Y-T, Park H-S. An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds. Mathematics. 2021; 9(18):2223. https://doi.org/10.3390/math9182223
Chicago/Turabian StyleKim, Yoon-Tae, and Hyun-Suk Park. 2021. "An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds" Mathematics 9, no. 18: 2223. https://doi.org/10.3390/math9182223
APA StyleKim, Y.-T., & Park, H.-S. (2021). An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds. Mathematics, 9(18), 2223. https://doi.org/10.3390/math9182223