## __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**