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