課程大綱

課程資訊

課程名稱

機率論一
Probability Theory (Ⅰ)

開課學期

106-1

授課對象

理學院應用數學科學研究所

授課教師

王振男

課號

MATH7509

課程識別碼

221 U3410

班次

學分

3.0

全/半年

半年

必/選修

必修

上課時間

星期二6,7(13:20~15:10)星期四7(14:20~15:10)

上課地點

天數101天數101

備註

總人數上限：33人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1061MATH7509_

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課程概述

This is a graduate course in measure-theoretic probability theory. We will begin with the review of basic measure theory suitable for the probability theory. Having reviewed measure theory, we can introduce random variables, random processes, distributions, and independence. We will discuss Laws of Large Numbers, Central Limit Theorems, Conditioning, Martingales, Markov processes, Random walks.

課程目標

Investigate some basic topics of random phenomena and learn the essential tools for studying such phenomena.

課程要求

1. Measure theory.
2. Undergraduate probability theory.

預期每週課後學習時數

Office Hours

另約時間備註： By appointment

指定閱讀

Foundations of Modern Probability, by Olav Kallenberg, second edition, Springer-Verlag, 2001.

參考書目

1. Probability: Theory and Examples, by R. Durrett. 4rd edition. Cambridge U.
Press 2010.
2. A Course in Probability Theory, by K.L. Chung, second edition, Academic Press, 1974.
3. Probability and Measure, by P. Billingsley, 3rd edition, Wiley, 1995.

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Homework	70%	weekly homework assignments
2.	Final	30%

課程進度

週次	日期	單元主題
第1週	9/12,9/14	\sigma-fields, Borel fields, \pi-\lambda system, Dynkin's theorem, Product spaces, Product \sigma-fields, Measurable functions.
第2週	9/19,9/21	Simple functions, Approximation of a positive measurable function by simple functions, Measures, Countable additivity, Integration, Monotone convergence theorem, Fatou's lemma, Integrable functions, Dominated Convergence theorem, Substitution, Density function, Completion (with subsets of measure zero sets).
第3週	9/26,9/28	Product measures, Fubini's and Tonelli's theorems, Probability kernels, Probability spaces, Random variables, Distribution or law of a random variable, Infinitely often, Ultimately, Borel-Cantelli's lemma (first part), \sigma-field on the set of functions, Processes, Paths, Finite-dimensional distributions, Distribution functions.
第4週	10/03,10/05	Convexity, Jensen's inequality, Covariance, Uncorrelated, Independence, Extension of independence to \sigma-field, Pairwise independence, P-trivial and degeneracy, Product measure.
第5週	10/10,10/12	Tail \sigma fields, Tail events, Kolmogorov's zero-one law, Convergence of random series and average.
第6週	10/17,10/19	Symmetric events, Hewitt-Savage zero-one law, Random walks, Borel-Cantelli Lemma, Uniform distribution, Bernoulli sequence, Version, Indistinguishability.
第7週	10/24,10/26	Moments and continuity (Kolmogorov, Loeve, Chentsov), Holder continuity, Distribution functions (one-dimension and multi-dimension), Moments and tails (Chebyshev's inequality), Almost sure convergence, Convergence in probability.
第8週	10/31,11/02	Cauchy in probability, Completeness, Weak convergence, Convergence in distribution, Tightness, Weak convergence and tightness, Tightness and convergence in probability, Uniform integrability.
第9週	11/07,11/09	Convergence of means, Convergence in L^p, Weak convergence in L^p, Weak L^1 compactness, Convergence of positive random series, Maximal inequality, Variance criterion.
第10週	11/14,11/16	Etemadi inequality, S_n converges a.s. iff S_n converges in probability.
第11週	11/21,11/23	Convergence of random series with independent symmetric terms, Characteristic function of a random variable, Median, Symmetrization.
第12週	11/28,11/30	Centering lemma, Proof of three-series criterion (Kolmogorov, Levy), Strong law of large number, Weak law of large number, Kronecker's lemma, Kolmogorov SLLN, Marcinkiewicz-Zygmund SLLN.
第13週	12/05,12/07	Proof of SLLN, The case when the expected value is infinite, Feller's theorem, Applications of SLLN, Empirical distribution functions, Bernstein's proof of Weierstrass' approximation theorem.
第14週	12/12,12/14	Characteristic functions, Fourier transform, Laplace transform, Generating functions, Properties of a characteristic function, Tail estimates, Tightness and equicontinuity, Characteristic functions and continuity, Characterization of a characteristic function, Non-negative definite functions, Bochner's theorem.
第15週	12/19,12/21	Poisson convergence, Null arrays, Poisson distribution, Central Limit Theorem.
第16週	12/26,12/28	Approximation by CLT, CLT for triangular arrays of symmetric random variables, Lindeberg-Feller CLT.
第17週	1/02,1/04	Proof of Lindeberg-Feller CLT, Lindeberg's condition, Feller's condition, Feller-Levy three series theorem, Weak Law of Large Numbers.