STAT 7010-0

Brief

The ob<x>jective of this course is to offer students a more profound comprehension of probability theory tailored for statistical research. This is a PhD level course whose prerequisite knowledge includes: mathematical analysis and mathematical statistics. Elementary real analysis is not necessary, but is better than none. Topics to be covered include: almost sure convergence, convergence in distribution, conditional expectation, martingale inequalities, central limit theorem for sums of independent random variables, for martingale difference sequences, for m-dependent random variables, and for U-statistics, stochastic regression theory, weak convergence, Donsker's theorem, and concentration inequalities.

Description

Course keywords: Almost sure convergence, Central limit theorem, Martingales, Weak convergence, Concentration inequalities, Brownian motion 一、課程說明(Course Description) The objective of this course is to offer students a more profound comprehension of probability theory tailored for statistical research. This is a PhD level course whose prerequisite knowledge includes: mathematical analysis and mathematical statistics. Elementary real analysis is not necessary, but is better than none. 二、指定用書(Text Books) 三、參考書籍(References) 1. Y. S. Chow and H. Teicher (1997). Probability Theory: Independence, Interchangeability, Martingales. 3rd Edition, Springer, Berlin. 2. P. J. Brockwell and R. A. Davis (1987). Time Series: Theory and Methods, Springer-Verlag. 3. P. Billingsley (1968). Convergence of Probability Measure. John Wiley & Sons, Inc. 4. P. Buhlmann and S.Van De Geer (2013).Statistics for High-Dimensional Data: Methods, Theory and Applications. Springer-Verlag. 5. C.Z. Wei's lecture notes in martingales. 6. D. Pollard (1984). Convergence of Stochastic Processes. Springer-Verlag. 四、教學方式(Teaching Method)(必填) 口述 五、教學進度(Syllabus)(必填) 1. Almost sure convergence 2. Convergence in distribution 3. Conditional expectation and martingale inequalities, 4. Central limit theorem for sums of independent random variables, for martingale difference sequences, for m-dependent random variables, and for U-statistics. 5. Stochastic regression theory 6. Concentration inequalities 7. Weak convergence, Brownian motion and Donsker's theorem, 六、成績考核(Evaluation)(必填) Homework 20% Midterm presentation 40% Final presentation 40% 七、 下列何項 AI 使用規則 (Indicate which of the following options you use to manage student use of the AI) 　有條件開放，請註明如何使用生成式AI於課程產出 Conditionally open; please specify how generative AI will be used in course output

Timetable

Scores

Average GPA 4.13

Std. Deviation 0.15

Remarks

本課程為 16 週課程

Restrictions

Compulsory

Elective

Cross Discipline

First Specialization

Second Specialization