Course Description
The focus of this course is to introduce the measure-theoretic formulation of the probability
theory to the students. I will start from the basic limit notions of set theory and then develop
measure-theoretic tools to define a probability measure, random variables, different kinds of
convergence, and end with central limit theorems. Results from the Real Analysis course will be
a prerequisite for this.
Course Content
I will try to cover the following topics in this course.
• Limits of a sequence of sets, Fields and sigma-fields, Probability measure, Probability space,
sigma-finite measures, Lebesgue measure, Carath´eodory extension theorem
• Measurable functions and integration, Properties of Borel measurable functions, Basic
integration theorems (Monotone convergence theorem, Extended monotone convergence
theorem, Fatou’s lemma, Dominated convergence theorem), Comparison of Lebesgue and
Riemann integrals
• Random variables, Induced probability space, Distribution functions, Expectation and
moments, Inequalities, Convergence of sequence of random variables (Almost surely, Probability
convergence, rth mean)
• Independence of events and random variables, Infinite sequence of random variables,
Conditional probability
• Convergence of sequence of distribution functions, Helly-Bray theorems, Convergence of
moments
• Convergence of series of independent random variables, Weak law of large numbers,
Kolmogorov’s strong law of large number, Kolmogorov’s zero-one law
• Central limit theorem, Convergence to a normal distribution, Lindeberg-Levy and Lindeberg
-Feller
Course Audience
At least one course in Real Analysis is mandatory for this course. Any student who satisfies this criterion may take this.
Outcomes of this Course
Grading
The total marks will be distributed over the two examinations (mid-term and end-term) and four quizzes.
There will be two quizzes before the mid-term and two after the mid-term. Each quiz will carry
ten marks, and both the midterm and end-term will have thirty credits each.
Assignments
I will give assignments every two weeks (tentatively). Students need to solve those as preparation
for the quizzes and the term exams. They are encouraged to try on their own. I will discuss the
problems in the tutorial class if necessary.
References
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K.L.Chung: A Course in Probability Theory, Third Edition, Academic Press, 2001.
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B.R.Bhat: Modern Probability Theory, Third Edition, New Age International (P) Ltd, 2004.
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M. Loéve: Probability Theory-I, Graduate Text in Mathematics, Fourth Edition, Springer, 1977.
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A.K. Basu: Measure Theory and Probability, PHI Learning Private Limited, 2012.
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Robert B. Ash, Catherine Doleans-Dade: Probability and Measure Theory, Harcourt Academic Press, 2000.