Lecture Notes

LEC # TOPICS NOTES
1

Probability Spaces, Properties of Probability

Caratheodory Extension Theorem, Approximation Lemma

(PDF)
2-3

Random Variables and their Properties

Independence

Expectation

(PDF)
4 Kolmogorov's Theorem about Consistent Distributions (PDF)
5 Laws of Large Numbers (PDF)
6

Bernstein's Polynomials, Moment Problem on [0, 1]

Hausdorff and de Finetti Theorems

Polya Urn Scheme

(PDF)
7

Kolmogorov and Savage-Hewitt 0-1 Laws

First Kolmogorov's Inequality

Convergence of Random Series

(PDF)
8

Stopping Times, Wald's Identity

Markov Property, Another Proof of SLLN

(PDF)
9-10

Convergence of Laws, Uniform Tightness and Selection Theorem on Metric Spaces

Stone-Daniell Theorem (no proof)

(PDF)
11

Characteristic Functions, Convolutions, Central Limit Theorem on the Real Line

Uniqueness Theorem

(PDF)
12

Fourier Inversion Formula

Multivariate Normal Distributions and Central Limit Theorem

Continuous Mapping Theorem

(PDF)
13

Lindeberg's CLT

Levy's Equivalence Theorem, Three Series Theorem

(PDF)
14

Levy's Continuity Theorem

Example of Poisson Approximation

Conditional Expectation

(PDF)
15-16

Martingales, Doob's Decomposition

Uniform Integrability and L_1 Convergence Theorem

(PDF)
17

Optional Stopping Theorem for Martingales

Doob's Maximal and Upcrossing Inequalities, Second Kolmogorov's Inequality

(PDF)
18-19

Convergence of (Sub)martingales

Levy Convergence Theorem

Applications: SLLN, Kolmogorov's LLN, Polya Urn Scheme

Fundamental Wald's Identity, Crossing a Growing Boundary

(PDF)
20

Convergence On Metric Spaces, Portmanteau Theorem

Lipschitz Functions, Extension Theorem

(PDF)
21 Stone-Weierstrass Theorem (PDF)
22

Metrics for Convergence of Laws: Levy-Prohorov, Bounded Lipschitz metrics

Ulam's Theorem, Convergence of Laws on Separable Metric Spaces

(PDF)
23

Ky Fan metric, Convergence and Uniform Tightness

Characterization of Uniform Tightness on Complete Separable Metric Spaces

Prohorov's Theorem

(PDF)
24-25 Close Laws and Close Random Variables: Strassen's Theorem (PDF)
26-27

Relationship Between Ky Fan and Levy-Prohorov Metrics

Wasserstein's Metric. Kantorovich-Rubinstein Theorems

(PDF)
28-29

Prekopa-Leindler Inequality

Entropy and Kullback-Leibler divergence

Transportation Inequality for Wasserstein Metric for Log-concave Measures, Concentration of Gaussian Measure

Trivial Metric and Total Variation

(PDF)
30

Stochastic Processes, Continuous Processes

Brownian Motion, Existence of Brownian Motion

Empirical Distributions and Kolmogorov-Smirnov Test

(PDF)
31 Uniform Tightness on the Space of Continuous Functions, Asymptotic Equicontinuity and Donsker Invariance Principle (PDF)
32-33 Equicontinuity of Empirical Process, Symmetrization, Hoeffding's Inequality, Kolmogorov's Chaining, Dudley's Entropy Integral (PDF)
34-35

Markov Property of Brownian Motion

Brownian Bridge as Pinned Down Brownian Motion, Reflection Principles, Laws of Suprema and the Spread

(PDF)
36

Laws of Brownian Motion at Stopping Times

Skorohod's Imbedding

(PDF)
37 Laws of the Iterated Logarithm (PDF)