解析與機率二   (88年度 下學期)

             N=新數館   Ns=新數館討論室   O=舊數館

Plan to cover Chpt.1- Chpt.9 on the first semester. We shall start introducing the measure space, probability space, integration, and expectation. (Chpt.1- Chpt.3, Chpt.5). Thus no need to learn theory of real variable or measure theory beforehand. The rest of the couse will focus on classical limit theorems of probability, which include: 1. various convergence concepts (Chpt.6), 2. weak and strong law of large numbers, three series theorems (Chpt.7), 3. convergence in distribution (Chpt.8), 4. moment generating function, characteristic functions, selection theorem, tightness of probability distributions, Prohorov's theorem, classical and Lindeberg-Feller central limit theorems. (Chpt.9). In the second semester we plan to continue with the last chpt. , chpt.10, of the text. Here we discuss the Radon-Nikodym derivatives, conditional expectations, discrete time martingales, stopping times, and martingale convergence theorem.A short survey of the relevance of martingales to math. Finance is given at the end of the text.Next we study ergodic theorems. After that we shall go to stochastic processes, including Poission process, Markov chains, and, if time is enough, Brownian Motions. These will be treated at an introductory level, and may change in response to audiances' interests.

Text: S. I. Resnick, A probability path Birkhauser, 1999.