MSc thesis / 2021
Stein’s Method, Malliavin Calculus, Relations and Applications
Connecting Stein’s method and Malliavin calculus to quantitative normal approximation for functionals of Gaussian processes.
- Authors
- Keivan Mirzaei
- Institution
- Sharif University of Technology
- Supervisor
- Prof. Bijan Zohuri-Zangeneh
- Assistant supervisor
- Dr. Mahdieh Tahmasebi
Overview
This thesis studies how Stein’s method and Malliavin calculus work together to quantify how closely functionals of Gaussian processes follow a normal distribution. It develops the Gaussian-analysis background, introduces the central operators of Malliavin calculus, and studies their connections with normal approximation.
Topics
- Gaussian analysis: Hermite polynomials, tensor products of Hilbert spaces, isonormal Gaussian processes, and Wiener chaos.
- Malliavin calculus: the derivative, divergence, and Ornstein-Uhlenbeck operators, together with multiple Wiener-Itô integrals and integration-by-parts formulas.
- Normal approximation: Berry-Esseen-type bounds and exact asymptotics for central limit theorems involving Gaussian functionals.
Applications
The applications include Toeplitz quadratic functionals of stationary Gaussian processes, quadratic functionals of the Brownian sheet, and refinements of Breuer-Major central limit results associated with fractional Brownian motion.
The document
The thesis was submitted at Sharif University of Technology and is dated March 2021. It is written in Persian and includes an English abstract and title page. The complete 121-page PDF is available through the link above.