Holooly Plus Logo
Holooly Plus Logo

Question 11.5.15: Show that any ergodic Markov chain with a symmetric transiti......

Show that any ergodic Markov chain with a symmetric transition matrix (i.e., pij = pji) is reversible.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.
Report Answer

If pij = pji then P has column sums 1. We have seen (Exercise 9 of Section 11.3) that in this case the fixed vector is a constant vector. Thus for any two states si and sj , wi = wj and pij = pji.Thus wipij = wj pji, and the chain is reversible.

Related Answered Questions

Question: 11.2.21

(Roberts7) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts u1, u2, and u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from region i ends up the next day at region j. A fraction qi =1−Σj qij > 0 goes into the atmosphere and ...

Verified Answer:

The problem should assume that a fraction q...
Question: 11.2.33

Consider a random walker who moves on the integers 0, 1, …, N, moving one step to ...

Verified Answer:

In each case Exercise 27 shows that f(i) = biNf(N)...
Question: 11.2.31

In Example 11.11, define f(i) to be the proportion of G genes in state i. Show that f is ...

Verified Answer:

You can easily check that the proportion of G’s in...
Question: 11.2.27

Consider an absorbing Markov chain with state space S. Let f be a function defined on S ...

Verified Answer:

Use the solution to Exercise 24 with w = f.
Question: 11.5.23

Assume that an ergodic Markov chain has states s1,s2,…,sk. Let Sj^(n) denote the number ...

Verified Answer:

Assume that the chain is started in state si. Let ...
Question: 11.5.21

A highly simplified game of “Monopoly” is played on a board with four squares as shown in ...

Verified Answer:

The transition matrix is P= \begin{matrix} ...
Question: 11.5.19

Show that, for an ergodic Markov chain (see Theorem 11.16), Σ j mijwj = Σj zjj -1 = K. ...

Verified Answer:

Recall that m_{ij}= \sum\limits_{j}{\frac{z...
Question: 11.5.18

A perpetual craps game goes on at Charley’s. Jones comes into Charley’s on an evening ...

Verified Answer:

Form a Markov chain whose states are the possible ...
Question: 11.5.17

An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector ...

Verified Answer:

We know that wZ = w. We also know that mki = (zii ...
Question: 11.5.13

Let P be the transition matrix of an ergodic Markov chain and P^∗ the reverse transition ...

Verified Answer:

Assume that w is a fixed vector for P. Then ...