QUICK SKETCH OF MAJOR TOPICS FOR STAT 650 FINAL, SPRING 2018
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A. Discrete-time Markov Chains --- Heavy emphasis
       State-space decompositions
	   Chapman-Kolmogoroff Equation
	   Transience, recurrence and pos-recurrence def'ns
	   Periodicity, etc.
	   Convergence Theorems
	   First-step analysis and exit distributions

	   
B. Continuous-time Markov chains
       Construction via holding times
	   stationary distributions 
	   detailed balance and reversibility
	   convergence theorems

	   
C. Poisson Process
       alternative equivalent definitions
	   calculations based on memoryless property of exponential
	   theorems about superposition and thinning
	   uniform distribution of jumps in an interval given their number

	   
D. Renewal Processes
       definitions
	   basic limit theorems related to SLLN
	   long-term averages and proportions based on cycles
	   Renewal Equations via first-step analysis
	   Existence and uniqueness of solutions to renewal equations
	   Renewal Theorems -- Basic, Blackwell, and Key
	   Relation between Markov Chain limit theorems and Renewal Theorems

	   
E. Martingales
       Definitions and optional stopping theorem 
	   Wald's Equation
	   Martingale Convergence Theorems
	   Application to recurrence and boundary-exit probabilities 
	        for (discrete-state) Markov Chains
	   
Specific Application Ideas: you should certainly know what Branching 
Processes and Birth-Death chains are, and what PASTA means and how 
it is applied in Queueing and Cycle problems.