Question

A transition probability matrix \(\mathbf{P}\) is said to be doubly stochastic if the sum over each column equals one; that is, $$ \sum_{i} P_{i j}=1, \quad \text { for all } j $$ If such a chain is irreducible and aperiodic and consists of \(M+1\) states \(0,1, \ldots, M\), show that the limiting probabilities are given by $$ \pi_{j}=\frac{1}{M+1}, \quad j=0,1, \ldots, M $$

Short answer

The limiting probabilities for the given irreducible, aperiodic, and doubly stochastic Markov chain with \(M+1\) states are \(\pi_j = \frac{1}{M+1}\) for \(j = 0,1, \ldots, M\).

Step-by-step solution

Understanding the properties of the given Markov chain

It is given that the transition probability matrix \(\mathbf{P}\) is doubly stochastic, which means the sum of probabilities in each column equals 1. Also, the chain is irreducible and aperiodic, implying that it is possible to reach any state from any state in a finite number of steps, and there is no fixed period for which the chain returns to the same state.

Writing the balance equations

In order to find the limiting probabilities \(\pi_j\), we need the balance equations, which are given by: $$ \pi_j = \sum_{i=0}^M \pi_i P_{ij}, \quad j = 0,1, \ldots, M $$

Proving that all limiting probabilities are equal

Since the matrix is doubly stochastic, we know that: $$ \sum_{i=0}^M P_{ij} = 1, \quad j = 0,1, \ldots, M $$ Substitute the balance equations into this equation: $$ \sum_{i=0}^M \pi_i P_{ij} = \sum_{i=0}^M \pi_i = 1, \quad j = 0,1, \ldots, M $$ Notice that the sum of the limiting probabilities is the same for each column, indicating that all the limiting probabilities are equal. To represent this, we can write: $$ \pi_j = \pi, \quad j = 0,1, \ldots, M $$

Finding the limiting probabilities

Since we know all limiting probabilities are equal, we can sum up all the probabilities and set them equal to 1. This will give us the value of \(\pi\): $$ (M+1) \pi = 1 $$ Solve for \(\pi\): $$ \pi = \frac{1}{M+1} $$ Therefore, the limiting probabilities are given by: $$ \pi_j = \frac{1}{M+1}, \quad j = 0,1, \ldots, M $$