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Lecture Notes of Probability Theory

This is a repository for materials written for the course on probability theory.

Our Progress

  • Week 0

    • Introduction

  • Week 1

    • Cardinality of sets
    • Ordinals
    • Axiom of choice
    • Tons of exercises

  • Week 2

    • Cantor set, normal number
    • How to define a measures on $\mathbb{R}$?
    • Baire Category Theorem
    • Existence of unmeasurable sets

  • Week 3

    • $\sigma$-algebra, $\pi$-system, Borel sets
    • $\sigma$-additivity and its consequence
    • Lebesgue measure, measurable sets,completion of measure space
    • Probability space, random variable
    • Carathéodory’s Extension Theorem

  • Week 4

    • Definitions: Event, $(E_n, \text{i.o.}), (E_n, \text{ev})$, almost sure(a.s.)
    • Borel-Cantelli lemma (BC1)
    • Poincaré’s Recurrence Theorem
    • Continuity of measure, Fatou’s Lemma and Reverse Fatou Lemma
    • Measurable function, $\sigma$-algebra generated by r.v.

  • Week 5

    • Doob–Dynkin Lemma
    • Independence

  • Week 6

    • Lovasz Local Lemma
    • Product space
    • Kolmogorov Existence Theorem (This guarantees the existence of random process.)
    • Borel's 0-1 Law
    • Kolmogorov's 0-1 Law
    • Hewitt-Savage 0-1 Law

  • Week 7

    • Definition: Lebesgue integration for $(m\Sigma)^+$
    • Monotone-Convergence Theorem(MON)
    • The Fatou Lemmas for functions
    • $\mu$-integrable functions: $\mathcal{L}^1(S,\Sigma, \mu)$
    • Linearity of integration

  • Week 8

    • Standard machine
    • Radon-Nikodym Theorem
    • Dominance-Convergence Theorem
    • Modes of convergence
    • Scheffe's Lemma

  • Week 9 (International Labor Day)

  • Week 10

    • Supporting hyperplane theorem and Jensen's inequality
      • Fatou lemmas from (infinite-dimensional) Jensen's inequality.
    • Monotonicity of $\mathcal{L}^p$: Lyapounov's inequality and truncation method
    • Hölder's inequality
    • $\mathcal{L}^p / \sim$ is a Banach space: Minkowski's inequality (triangle inequality) and completeness
    • $\mathcal{L}^2$ is a Hilbert space: orthogonal projection
    • Covariance matrix as a Gram matrix of an inner product space

  • Week 11

    • Calculate expectation from law
    • Probability density function(pdf)
    • Gaussian distribution
    • An introduction of Central Limit Theorem(CLT) and Berry-Esseen theorem
    • Bernstein polynomials and Weierstrass approximation theorem

  • Week 12

    • Definition and existence of conditional expectation: Fundamental Theorem (Kolmogorov)
      • proof from Radon-Nikodym theorem
      • proof as least-squares-best predicator
    • Properties of conditional expectation
    • Definitions: filtration, adapted process, martingale
      • e.g. Doob martingale
    • Previsible process, martingale transform and (discrete) stochastic integral

  • Week 13

    • Simple examples of martingale
      • Simple random walk: sum (or product) of I.I.D random variables
      • "ABRACAADABRA" (c.f. E10.6. of the textbook)
      • Make a submartingale from a martingale with Jensen's inequality
    • Stopping time (or Optional time)
    • A characterization of martingale using stopping time

  • Week 14

    • More examples: Polya's Urn, branching process(c.f. Chapter 0), quadratic variation
    • Doob's maximal inequalities:
      • Doob's submartingale inequality
      • Doob's $\mathcal{L}^p$ inequality
    • Upcrossings and Doob's upcrossing lemma
    • Doob's 'forward' convergence theorem
      • e.g. 1D drunkard's walk
    • Doob decomposition
    • Uniform integrability
      • UI property: bridge between different modes of convergence
      • Sufficient conditions for UI property
      • UI property of conditional expectations

  • Week 15

    • Bounded Convergence Theorem
    • Convergence in probability + UI property $\iff \mathcal{L}^1$ convergence
    • Lévy's 'Upward' Theorem
    • Lévy's 0-1 law
    • Martingale proof of Kolmogorov's 0-1 law

  • Week 16

    • Radon-Nikodym derivative revisited
      • ${\frac{dP_n}{dQ_n}}$ is a martingale
      • Conditional expectation of $\frac{dP}{dQ}$
      • Calculate Radon-Nikodym derivative from pdf
      • Martingale proof of Radon-Nikodym Theorem (Meyer 1966)
    • Back to starting point: various interpretations of probability
      • Linear expectation
      • Sublinear expectation and its connection with linear expectation
      • Quantum expectation
      • An example from high-dimensional probability
    • Subadditive functions and rooted trees