Math 447 – Winter 2023 Assignment 2
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Math 447 – Winter 2023
Assignment 2
Due January 31, 2023
Some of these problems are adapted from Dobrow. For those problems, the numbers refer to the problem numbers in the book.
1. Give an example of a finite Markov chain satisfying the following con- ditions.
(a) There are 3 communicating classes.
(b) Each communicating class has a di↵erent period.
(c) The transition matrix P has Pu,u < 1 for all u 2 V .
(d) There is at least one vertex v such that u ! v for all u 2 V . (In other words, v is accessible from any state of the chain.)
Explain your example.
2. (3.8) Let
Consider a Markov chains on four states whose transition matrix P is
the block matrix
(a) Does the Markov chain have a unique stationary distribution? If
so, find it.
(b) Does limn!1 Pn exist? If so, find it.
(c) Does the Markov chain have a limiting distribution? If so, find it.
Explain your answers.
3. (3.29) Consider the Markov chain with transition matrix
(a) Identify the communication classes.
(b) Classify the states as recurrent or transient, and determine the
period of each state.
4. Let P = (Pu,v)u,v2V be a transition matrix. Let (Xn,n ≥ 0) and (Yn,n ≥ 0) be independent Markov chains, both with transition matrix P .
(a) Show that ((Xn,Yn),n ≥ 0) is a Markov chain with state space
V ⇥ V and show that the entries of the transition matrix Q =
(Q(u,u\ ),(v\ ,v\ ))(u,u\ ),(v,v\ )2V ⇥V are given by Q(u,u\ ),(v,v\ ) = Pu,vPu\ ,v\ .
(b) Show that if P is regular then Q is regular.
(c) Give an example where P is irreducible but Q is not irreducible.
2023-02-04