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Sections: 6.1 - 6.6, 6.14
Homogeneous Markov Chain
n-step Transition Matrix
Null and positive recurrent
Total Variation Distance (3 definitions)
Coupling of Two Markov Chains
Random Walk on a Graph
Theorems: 6.1.5, 6.1.8, 6.2.3, 6.2.4, 6.2.5, 6.2.9, 6.3.2, 6.3.3, 6.3.4, 6.3.5, 6.4.3, 6.4.6, 6.4.17, 6.5.4, 6.6.1, 6.14.9, Coupling Collison Time Bound on Mixing Time,
What is the mixing time for a random walk on the complete graph with self loops?
Give upper and lower bounds for the mixing time of a random walk on two complete graphs of size n connected by a single edge.
Can a single Markov Chain have more than one stationary distribution? Why or why not?
Consider a two state MC with states A, B. p(A,A) =1/2, p(A, B) =1/2. P(B,B) =1/4, P(B,A) =3/4. What is the stationary distribution? What are the eigenvalues of the adjacency matrix?
Is a branching process a reversible Markov Chain? How about SRW?
Decompose the state space of the branching process MC.
Prove that 3d random walk is transient.
Prove that 2d random walk is null recurrent.
What is the stationary distribution of the random walk on the complete bipartite graph where the left partition has n vertices, the right 2n, and at each step there is probability 1/2 of remaining in the current state.
Give a graph whose random walk has period 3.
Give a Markov chain whose state space is not irreducible.
Give a Markov chain whose state space can be partitioned into two closed sets of states.
Consider m balls lying in n bins. At each step we pick a ball at random out of a bin, and throw it into a bin chosen uniformly at random. Is this a Markov Chain? What is the state space? Clasify the chain. What is the stationary distribution? Give upper and lower bounds for its mixing time.