1. Introduction and Contents

Considering a parameterized nonlinear transformation, which may arise from partial differential equations or time-frequency analysis, when subjected to random input, yields an output that can manifest as a temporal-spatial or time-scale random field. The primary objective of this course is to determine the large-scale limit of this output field. Additionally, the course aims to quantify the discrepancy between the output and its large-scale limit. We expect students to gain a deeper understanding of the convergence speed of standardized partial sums of correlated random variables through this training. To achieve this goal, the content of this course will contain the spectral representation of random processes, the Stein method, the non-Stein method, and stochastic calculus.

2. Course Outline

We will guide the students in understanding stochastic analysis tools used to determine the convergence speed of the distributions of normalized partial sums/integrals of correlated

random variables/fields. This includes exploring the spectral representation of random processes, the Stein method, the non-Stein method, and fundamental formulas in stochastic calculus. Once students have grasped these basics, we will compare the convergence speeds obtained through the Stein and non-Stein methods. We expect that students will apply this acquired knowledge to investigate corresponding convergence rate problems in various research fields.

3. Prerequisites

linear algebra, undergraduate probability theory

4. Grading Scheme

Student presentation

5. Course Goal

(1) Understand the concept of Stein’s method.

(2) Apply Stein’s method to get the rate of convergence of the classical Central Limit Theorem (CLT).

(3) Prove the CLT for weakly dependent random variables.

(4) Introduce the multidimensional Stein method.

(5) Explore the CLT and its convergence speed for random vectors.

6. Reference material (textbooks)

Reference:

(1) Chatterjee, S. A short survey of Stein's method. arXiv preprint arXiv:1404.1392 (2014).

(2) Chen, L.H., Goldstein, L. and Shao, Q.M., 2011. Normal approximation by Stein's method (Vol. 2). Berlin: Springer.

(3) Krylov, N. V. (2002). Introduction to the theory of random processes, volume 43. American Mathematical Soc.

(4) Nourdin, I. and Peccati, G. (2012). Normal approximations with Malliavin calculus: from Stein’s method to universality. Number 192. Cambridge University Press.

7. Credit: 2

8. Course Number/ ID

No.: NCTS 5056 (三校聯盟之學生於課程網選課適用)

ID: V41 U4110