Class Hours: TR.
Text:
Probability Essentials by Jean Jacod and Philip Protter, Springer; 2nd edition.
Additional reading:
- Lecture notes on selected topics
- Probability: Theory and Examples
by Richard Durrett, Duxbury Press; 3 edition.
- Probability and Measure,
by Patrick Billingsley, Wiley-Interscience; 3rd edition.
Grading policy: Based on homework assigned about every two weeks plus two exams (at least one take home):
Description:
MTH721 - Probability theory - is a self-contained measure-theoretical probability course. It is the first semester
in a sequence of three graduate level courses on probability.
It covers rigorously the concepts of probability space,
random variable, distribution, independence,
the limit theorems (Law of Large Numbers and Central Limit Theorem) and
continues with discrete time martingales. The limit laws are the basis of
classical statistical inference and are a requisite for any mathematician.
Prerequisites:
The course is as self contained as possible.
MTH 224, 524/624, 510, 512, 534/634
recommended.
MTH 712, 725, 730-631 are helpful but not mandatory.
A natural sequel is MTH738
- Stochastic processes -
which starts with Markov chains (classification of
states, ergodic properties) and continues
with continuous time processes, makes the connection
with differential equations, and ends with the concept of scaling,
which leads in a natural way to Brownian motion.
This is the first step in
the study of diffusion theory via Ito calculus, a third semester. The area is
strongly related to partial differential equations.
- Measurable spaces. Sigma-fields, events, measurable
functions.
- Integrability. Product spaces.
Fubini's theorem.
- Probability space. Examples. Independence of events.
- Conditional probability. Tail Events. Borel-Cantelli lemma.
- Random variables. Distribution functions.
- Expectation. One-dimensional
distribution functions and positive measures.
- Transformations of
random variables.
- Examples. Discrete, continuous and mixed random variables.
- Multidimensional random variables. Correlation.
- Convergence. Kolmogorov, Chebyshev's inequalities. Types of
convergence. Tightness.
- Characteristic function. Properties. Moment generating functions.
- Sums of random variables. Convolution. Examples.
- Law of large numbers (LLN). Weak LLN. Strong LLN.
- Central Limit Theorem (CLT).
- Martingales. Convergence, inequalities. Martingale
decomposition. Examples.