STAT 650 SYLLABUS / COURSE OUTLINE
This syllabus is a revised version after March 12, 2020,
when changes to the UMD Spring Academic Schedule were first announced following the Coronavirus outbreak.
0. Probability Review.
(Durrett Appendix A, Lefebvre, Chapter 1; Serfozo, Sec.1.22 & Ch.6.)
2 Lectures
(a) Probability spaces, countable additivity.
(b) Conditional expectation, mixed-type joint densities
1. General Definitions and Properties of Stochastic Processes
(Class Notes and Lefebvre, Chapter 2.) 2 Lectures
(a) Spaces of trajectories, Kolmogorov Consistency Theorem (infinite sequences of states).
(b) Definition of Infinite Proces Trajectory on Infinite Product Space; Simulation
2. Discrete-time Discrete-State Markov Chains.
(Serfozo, Chapter 1 and Durrett Chapter 1) 8 Lectures
(a) Markov property. Examples of Markov & non-Markov random sequences.
(b) Multistep transition probabilities. Chapman-Kolmogorov equation.
(c) "First-step analysis" and branching processes
(d) Classification of states.
(e) Notions of limiting behavior. Reducibility. Recurrence. Steady state.
(f) Time reversibility, exit distributions, and other topics.
3. Renewal Processes.
(Serfozo Chapter 2, and Durrett Chapter 3) 2 Lectures
(a) Renewal Counting Process, Renewal Function
(b) Laws of Large Numbers, Renewal Equation.
4. Poisson Processes.
(Serfozo Chapter 3, and Durrett Chapter 2) 5 Lectures
(a) Memoryless Property of Exponential. Independent & Stationary Increments Processes.
(b) Three Equivalent Definitions of Poisson Process
(c) Nonhomogeneous Poisson, Compound Poisson, Superposition and Thinning
5. Continuous time Markov Chains.
(Serfozo Chapter 4, and Durrett Chapter 4) 5 Lectures
(a) Kolmogorov Forward and backward equations. Birth-death processes.
(b) Embedded discrete chains.
(c) Limiting behavior and stationary distributions.
6. Martingales & Applications.
(Serfozo Sections 5.5--5.7, and Durrett Chapter 5) Maybe 1 Lecture
(a) Definitions, Optional Sampling Theorem, Expectation calculations.
7. Application: Markov Chain Monte Carlo.
(Serfozo Sections 1.18 plus Notes.) 1 Lecture