Probability II

Department: 
MATH
Course Number: 
6242
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every spring semester

Develops the probability basis requisite in modern statistical theories and stochastic processes. (2nd of two courses)

Prerequisites: 

MATH 6241 or equivalent

Course Text: 

At the level of Billingsley:  Probability and Measure

Topic Outline: 
  • Characterizations and limit theorems for sums of independent random variables, such as extensions and/or error analysis for central limit theorems, the iterated law of large numbers, properties of infinitely divisible distributions, and tail events, symmetric events and zero-one laws
  • Random walks, including basic properties and recurrence results
  • Conditional probability and conditional expectation, including basic properties and connections to Radon-Nikodym derivatives, projections, etc.
  • Markov processes, including basic properties and examples, stopping times and the strong Markov property, use of transition probabilities, and applications
  • Martingales, including basic inequalities and convergence theorems, optional sampling, backward martingales, and applications
  • Ergodic theory, including basic definitions and examples, and topics such as the Pointwise Ergodic Theorem, recurrence and mixing
  • Poisson processes and Brownian motion, including basic constructions and some basic properties