Course Information

  • Time: Mon 7-10 pm
  • Location: 三教 107
  • Office hour: M622, Mon 4-6 pm or by appointment
  • TA: 樊泽腾 (fanzet at hotmail dot com)

Lecture note

lecture note (last update: 12/26 )

hand-written draft notes

datefile
9/14lect 2
9/23lect 3
9/30lect 4
10/14lect 5
10/21lect 6
10/28lect 7
11/4lect 8
11/11See the lecture note
12/9lect 11

references

  • Durrett, Richard. Probability: Theory and Examples (Fifth edition). Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019. Copy from Durrett’s homepage
  • Chung, Kai Lai. A Course in Probability Theory (Third edition). San Diego, Calif.: Academic Press, 2007.
  • Billingsley, Patrick. Convergence of Probability Measures (2nd edition). Wiley Series in Probability and Statistics. Probability and Statistics. New York: Wiley, 1999.
  • Kolmogorov, A.N. Foundations of the Theory of Probability (English Translation). Edited by Nathan Morrison. 1st ed., 1933. PDF copy
  • Shiryaev, A. N. Probability. Vol. 95. Graduate Texts in Mathematics. New York, NY: Springer New York, 1996. Copy from Springer.

HW

  • weekly; posted usually by Wednesday, due next Monday before class.
  • Discussion and collaboration are encouraged, but names of the collaborators should be given in the submitted work.
AssignmentsDue date
HW19/23
HW29/30
HW310/14
HW410/21
HW510/28
HW611/11
HW711/18
HW811/25
HW912/2
HW1012/16
HW1112/23

Grading scheme

%
HW assignments40%weekly
Final60%

Schedule (tentative)

  • Measure theory preliminaries (weeks 1-2): properties of measures, probability spaces; sigma-algebras, monotone class theorem, principle of appropriate sets; distribution functions, construction of measures, Carathéodory’s extension theorem; decomposition of distribution functions, singular and absolute continuity, singular measures, Cantor set and function.
  • Random variables and integration (week 2): measurable maps, random variables and vectors, Borel measurability; simple functions, expectation of random variables; monotone/dominated/bounded convergence theorems, Fatou’s lemma; change of variables.
  • Mode of convergence (week 3): almost sure convergence, convergence in probability, convergence in Lp, convergence in distribution; first Borel–Cantelli lemma; uniform integrability.
  • Independence and product measures (week 4): independence of r.v.’s, independence of sigma-algebras; product measures, Fubini’s theorem, cross-section, Fubini on complete measure spaces; Kolmogorov’s extension theorem, cylinder sets, consistency condition, iid r.v.’s.
  • Law of large numbers (weeks 5-6): uncorrelated r.v.’s, Chebyshev’s inequality, L2 LLN; weak LLN, convergence of triangular arrays; second Borel–Cantelli lemma, strong LLN, tail sigma-algebras, Kolmogorov’s 0-1 law, Kolmogorov’s three-series theorem.
  • Central Limit Theorem (weeks 7-8): weak convergence; characteristic functions and inversion formula; Levy’s continuity Theorem; Lindeburg’s condition; stable laws and infinitely divisible distributions
  • Discrete martingales (weeks 9-10): conditional expectations; Doob’s inequalities, uniformly integrable martingales, down-crossing inequality and a.s. convergence; stopping times, optional sampling theorem
  • Markov chains (weeks 11-12): random walks, zero-one law, recurrence; stationary measures; strong Markov property.
  • Ergodic Theorems (week 13): measure preserving map, invariant sets; invariant measure, ergodic measure; Birkhoff’s Ergodic Theorem; subadditive ergodic theorem.
  • Functional CLT (week 14): measures on metric spaces, weak convergence; Brownian motions; Donsker’s invariance principle; CLT for martingales; CLT for stationary sequences; mixing.
  • Other topics and review (weeks 15-16): regular conditional expectation; renewal process